In the Srimad Bhagavatam it is stated,

anye ca samskrtatmano
vidhina abhihitena te
yajanti tvan mayas tvamvai
bahumurty ekamurtikam

“Lord Krishna says: There are others, who are purified and cultured souls, who by understanding my different forms as myself, as one personality in various forms and features, worship me according to their spiritual understanding and convictions following the rules of worship as given in the Vedas.”

Each process of worship given in the vedic scriptures involves understanding the different forms of the Lord, but the ultimate purpose is to worship the same one Supreme Lord Himself.

The various forms of the Lord which are unlimited but one and the same are described in the vedic literatures as follows:

Sri Krishna , the Supreme Personality of Godhead manifests Himself in three features in His transcendental form:

svayam rupa – His personal form,

tad ekatma rupa – different form but Himself,

avesa rupa – form into which He has entered.

The svayam rupa is the form in which Sri Krishna is directly understood, His personal form.

The tad ekatma rupa is that form which most resembles the svayam rupa, but there are some differences in the bodily features.

The avesa rupa is when Krishna enters either his own expansions or some suitable qualified living entity to represent Himself in any one or more of His opulences or powers. This is also known as shaktyavesha incarnation.

Vasudeva, the identity by which Lord Sri Krishna is known as the original Personality of Godhead, thus expands Himself by innumerable forms, which are all transcendental to matter, known as sat (eternally existent), chit (fully conscious) and ananda (fully blissful).

The personal form of Krishna can be divided into two: svayam rupa and svayam prakasa.

As far as His svayam rupa or pastime form is concerned , it is in this form that He remains always in His personal abode, Vrindavana, with the inhabitants of Vrindavana (vraja vasis).

This svayam rupa form can be further divided into prabhava and vaibhava forms:

(a) When Krishna expanded Himself into multiple forms during the rasa dance in order to dance with each and every gopi who took part in the dance. (prabhava expansions)

(b) At Dwaraka Krishna expanded Himself into 16,108 forms in order to accommodate His 16,108 wives. Each such expansion was a separate individual, unlike when some yogis expand themselves into eight forms where all of them act as one individual (if one walks the others have to, if one sits the others also sit etc).

When Narada (who is himself a yogi, who can do such tricks like expanding bodily features into eight forms) visited Dwaraka, he was so astonished to see that in each of the 16,108 palaces , Sri Krishna was acting in different ways with His queens. I one palace He was engaged in playing with His children, and in yet another form He was performing some household work. (vaibhava expansions)

Each of these categories can also be further divided into prabhava prakasha (pastime manifestation forms), prabhava vilasa (pastime extension forms), vaibhava prakasha (emotional manifestation forms) and vaibhava vilasa (emotional extension forms):

a) When Akrura was accompanying both Krishna and Balarama to Mathura from Gokula, he entered into the waters of the Yamuna river and saw within the waters of Yamuna all the planets of the spiritual realm. He saw the Vishnu form as well as Narada and the four Kumaras who were worshiping him. This is a prabhava prakasha category of expansion.

b) When Krishna appeared before His mother Devaki in His four handed form of Narayana just before His divine birth in Mathura, He appeared in His prabhava prakasha form.

c) When he changed Himself to a two handed form at the request of His parents, He appeared in His vaibhava prakasha form.

d) In His personal form, Krishna is just like a cowherd boy and He thinks of Himself in that way. This is a vaibhava prakasha.

e) When He is in Mathura as a kshatriya prince He thinks and acts as a princely administrator. This is a vaibhava vilasa form. (In this Vasudeva form, sometimes He gets attracted to His own cowherd boy form, as described in some vaishnava literatures).

The Lord performs many wonderful pastimes in His incarnations in the material realm. The different aspects of these pastimes are each a manifestation of certain qualities of the different categories of his expansions in the spiritual realm. Thus many categories of expansions of the Lord may manifest themselves within a particular incarnation in order to perform certain pastimes. These pastimes exhibit the distinct qualtities of that category of expansion.

There are also forms of Krishna which are a little different from the svayam rupa category. These are known as tad eka atma rupa forms (same person with different features).

These are further classified into vilasa (expansion form) and svamsha (own portion) forms.

Due to the distinct features both these classifications can be further divided into prabhava and vaibhava forms.

When Krishna expands Himself into Vasudeva, Pradyumna, Aniruddha and Sankarshana He appears as prabhava vilasa, under tad eka atma rupa category.

From these original principal four handed forms there are twenty four principal forms manifest, which are vaibhava vilasa under tad ekatma rupa category.

In the Siddhartha samhita the twenty four forms are named according to the position of symbols (viz, conch, disc, mace and lotus) in Their four hands.

These twenty-four forms are known as prabhava vilasa forms identified by the symbols held in Their four hands.

The four principal manifestations of Krishna are found in each planet in the spiritual realm, and these planets are called Narayanaloka or Vaikunthaloka.

In the Vaikuntha loka He is manifested in the four-handed form of Narayana. From each Narayana the forms of Vasudeva, Sankarshana, Pradyumna and Aniruddha are manifested.

The Narayana is the center and the four forms of Vasudeva , Sankarsana, Pradyumna and Aniruddha surround the Narayana form.

Each one of these four forms again expand into three:
Vasudeva – Kesava, Narayana, Madhava

Sankarshana – Govinda*, Vishnu, Madhusudana

Pradyumna – Trivikrama, Vamana, Sridhara

Aniruddha – Hrishikesha, Padmanabha, Damodhara
[*This Govinda is a four handed form different from the vaibhava prakasha Govinda, son of Nanda Maharaja, in Vrindavana.]

The four principal manifestations also have their vilasa (expansion) forms:
Vasudeva – Adhoksaja, Purusottama

Sankarshana – Upendra, Acyuta

Pradyumna – Nrisimha, Janardhana

Aniruddha – Hari, Krishna*
[*This Krishna is four handed vilasa form, different from the original svayam rupa form of Krishna.]

In the Hayasirsha Pancharatra it is mentioned that there are nine other forms protecting the Mathura and Dvaraka portions: Vasudeva, Sankarsana, Pradyumna and Aniruddha protect Mathura; Narayana, Nrsmha, Hayagriva, Varaha and Brahma (this is a four handed form of Krishna different from the creator Brahma who is a jiva) protect Dvaraka. These nine forms are different manifestations of prakasa and vilasa forms of Lord Krishna, distinct from the expansions with same names mentioned before in the tables above.

In the spiritual realm all the planets dominated by the Narayana features are eternal. The topmost planet is called Krishnaloka (Goloka) and is divided into three portions: Gokula, Mathura and Dvaraka.

The conclusion is that the Supreme Original Personality of Godhead is Krishna. He is called lila-purushottama (playful perfect person) and He resides principally as the son of Nanda. He has innumerable forms and categorically they are described in various scriptures especially the Pancharatra scriptures.

[About Vedic, Pancharatric and Tantric scriptures refer to the course on Vedic literature.]

All of these forms have their Vaikunta planets in the spiritual realm and are situated in eight different directions. Although all of them are ever situated in the spiritual realm, some of them are nonetheless manifest in the material world also.

The different categories of expansions in the Spiritual world are simply to facilitate the relationships with his separated expansions (the living beings in the spiritual realm) through various mellows.

To cite a few examples:

The svayam rupa form (the simple cowherd boy form) of the Lord is to enhance the parental affection of Nanda and Yasoda who are His eternal parents in the Vrindavana portion of Goloka.

The prabhava expansions of the Lord in the rasa lila are to enhance the conjugal satisfaction of the gopis with whom He dances.

The vaibhava expansion of the Lord in Mathura as a prince is to enhance the parental affection of Devaki and Vasudeva.

Some of the Lord’s forms expand throughout the material realm as deity incarnations (archa-avatar) to protect various areas. These deity incarnations are manifested to give facility to the devotees of the Lord so that they can worship Him in that particular deity form to whom they have developed attraction.

For example, Hayasirsa in Bhadrasva-varsa, Matsya in Ramyaka-varsa, Kesava in Mathura and Padmanabha in Anandaranya (Trivandrum).

In the Mathura portion of the Goloka (the topmost abode of Krishna in the spiritual realm) a form known as Kesava is situated. This form is also represented in earthly Mathura. Similarly there is a Purusottama form in the spiritual realm at the outskirts of Goloka which is represented in Jagannatha Puri on our earth. Some other examples of this kind of representation are the Padmanabha in Anandaranya (Trivandrum of Kerala in India), and Hari in Mayapur (near Calcutta in West Bengal in India) which is the birth place of Lord Sri Chaitanya.

Many other forms are also situated in various places on earth. Not only in this Universe but in all other universes as well, the forms of Krishna are distributed everywhere.

It is indicated that this earth is divided into seven islands, which are the seven continents, and it is understood that on each and every island there are similar forms, but in the recent past these seem to be present only in India. From the vedic literatures we can understand that there are forms in other parts of the world also.

Through the activities of the worldwide Hare Krishna Movement, many of these places are very recently retraced and the worship of various forms of Lord Sri Krishna are re-established on almost all the continents of the earth.

The different forms of Krishna which are distributed thus throughout the universe are there to give pleasure to the devotees. It is not that devotees are only in the eastern world or in India. There are devotees in all parts of the world, but at times they simply forget their identity. These forms manifest not only to give pleasure to the devotees but to reestablish the constitutional devotional service and perform other activities which vitally concern the Supreme Personality of Godhead.

yada yada hi dharmasya
glanir bhavati bharata
abhyutthanam adharmasya
tadatmanam srijamy aham

“Whenever and wherever there is a decline in religious practice, O descendant of Bharata, and a predominant rise of irreligion – at that time I descend Myself.”

paritranaya sadhunam
vinashaya ca duskritam
sambhavami yuge yuge

“To deliver the pious and to annihilate the miscreants, as well as to reestablish the principles of religion, I Myself appear, millennium after millennium.”

janma karma ca me divyam
evam yo vetti tattvah
tyaktva deham punar janma
naiti mam eti s’orjuna

“One who knows the transcendental nature of My appearance and activities does not, upon leaving the body, take his birth again in this material world, but attains My eternal abode, O Arjuna.” – Bhagavad-gita 4.7-9

The Brahma samhita (5.46) says:

diparcir eva hi dasantaram abhyupetya
dipayate vivrta-hetu-samana-dharma
yas tadrg eva hi ca visnutaya vibhati
govindam adi-purusam tam aham bhajami

“The light of one candle being communicated to other candles, although it burns separately in them, is the same in its quality. I adore the primeval Lord Govinda who exhibits Himself equally in the same mobile manner in His various manifestations.”

Such expansion of forms are made possible by His various energies. His energies are also multifarious, and His internal energies are superior and external energies are inferior in quality.

As explained in Bhagavad Gita (7.4-6):

bhumir apo ‘nalo vayuh
kham mano buddhir eva ca
ahankara itiyam me
bhinna prakritir astadha

“Earth, water, fire, air, ether, mind, intelligence and false ego – all together these eight constitute My separated material energies.”

apareyam itas tv anyam
prakritim viddhi me param
jiva-bhutam maha-baho
yayedam dharyate jagat

“Besides these, O mighty-armed Arjuna, there is another, superior energy of Mine, which comprises the living entities who are exploiting the resources of this material, inferior nature.”

etad-yonini bhutani
sarvanity upadharaya
aham krtsnasya jagatah
prabhavah pralayas tatha

“All created beings have their source in these two natures. Of all that is material and all that is spiritual in this world, know for certain that I am both the origin and the dissolution.”

Thus His expansions of various forms which take place via His internal energies are superior forms whereas the expansions which take place via the external energies are inferior forms.

The living entities are also His expansions. The living entities who are expanded by His internal potency are eternally liberated persons, whereas those who are expanded in terms of the material energies are eternally conditioned souls.

In a prison house, there are the prisoners who are undergoing punishment and the managers who staff the prison house. Even though both these categories are bound by the laws of prison house ordained by the king, the prisoners have no freedom and are there for punishment (the conditioned souls in the material world, expanded in terms of external energy).

The managers have freedom, they are there simply to make sure that the law and order is maintained (the invisible controllers of the material world, known as demigods. Also expanded in terms of external energy).

Sometimes a missionary comes to preach to the prisoners, he also has to follow a certain code of behavior under the laws of the prison, but he is free and he tries to correct the prisoners so that they can become free or at least behave better so that they wont suffer too much in the prison (the eternally liberated souls, expanded in terms of internal energy, who come to deliver the conditioned souls from the material creation are like these missionaries).

But the king himself sometimes visits the prison house. He is never bound by the laws of the prison, neither he needs to follow any code of conduct while visiting the prison. He is totally above all the laws of prison, and simply he visits so that everything goes on properly as ordained by his law. (The Lord is like the king, always transcendental to the laws of the material world).

The Lord is all spiritual and has nothing to do with the qualities of the material world, which are qualitatively distinct from the all spiritual qualities of the Lord. Because the causal and effectual energies required for the creation of the material world are also His energies, fully under His control, He is never affected by the material modes of nature. His position is always transcendental even within the material manifestation.

He is also superior and transcendental to the eternally liberated souls who are expanded by His internal energy because He is the Purusa uttama (Perfect Original Person, source of both internal and external energies):

dvav imau purushau loke
ksaras caksara eva ca
ksarah sarvani bhutani
kuta-stho ‘ksara ucyate

“There are two classes of beings, the fallible and the infallible. In the material world every living being is fallible, and in the spiritual world every living entity is called infallible.” – Bhagavad-gita 15.16

yasmat ksaram atito ‘ham
aksarad api cottamah
ato ‘smi loke vede ca
prathitah purusottamah

“Because I am transcendental, beyond both the fallible and the infallible, and because I am the greatest, I am celebrated both in the world and in the Vedas as the Supreme Person.” – Bhagavad-gita 15.18

The living entities are separated parts and parcels of the Lord , and the conditioned living entities, who are unfit for the spiritual realm, are strewn within the material world to fulfill their desire to enjoy matter to the fullest extent.

As the eternal friend of the living entities, the Lord expands as Paramatama the Supersoul (one of His plenary portions). Accompanying the living entities to guide them in their pursuit of material enjoyment and to become witness to all their activities, so that they can reap suitable reactions for their material activities in their multiple lives in matter.

The Mundaka Upanishad 3.1.1 describes the soul and Supersoul as two birds in a tree. One eating the fruit of the tree while the other is just witnessing the actions.

cva suparna sayuja sakhaya samanam vrksam parisasvajate
tayor anyan papplam svadv atty anasnann anyo ‘ abhicakasita

Thus the conditioned soul is controlled by the laws of material nature, but the Supersoul is the controller of both the spiritual and material energies.

Thus Vasudeva, the original Personality of Godhead by one of His plenary parts expands Himself all over the material world not only by accompanying the conditioned soul in all lives in different material bodies (numbering in 8,400,000 species), but also within the atoms, which combine to form the material bodies and the whole material universe. Matter, antimatter, proton, neutron etc are all simply different effects of this Supersoul, or Paramatma feature of the Lord.

The gross material bodies of the living entities are combination (in different proportions) of five gross elements (the atoms which are pervaded by His presence within them) and the subtle bodes consist of mind , intelligence and false identity (which are directed by His presence as Supersoul , Paramatma).

Bhagavad gita confirms that the Lord is the father of all living entities who are conceived within the womb of the material nature:

sarva-yonisu kaunteya
murtayah sambhavanti yah
tasam brahma mahad yonir
aham bija pradhah pita

“It should be understood that all species of life, O son of Kunti, are made possible by birth in this material nature, and that I am the seed-giving father.” – Bhagavad-gita 14.4

Since the living beings are His sons, the sufferings and enjoyments of the sons are also indirectly the sufferings and enjoyments of the father, even though the Father is not anyway directly affected. Being so kind He accompanies the living being as Supersoul always trying to divert the attention of them towards their constitutional nature, which is one of real happiness, ananda (bliss).

There are innumerable material universes, and in each and every universe there are innumerable planets inhabited by different grades of living entities in different modes of nature, in 8,400,000 species of bodies. The Lord incarnates Himself in each and every one of them and in each and every type of living species. He manifests His transcendental pastimes amongst them just to create a desire in them to go back to the spiritual realm, back home, back to Godhead.

This science of incarnations, their appearance and disappearance is nicely described in the Gopala Tapani Upanishad as follows:

avirbhava tirobhava svapade
tisthati tamasi rajasi satviki
manusi vijnanadhana anandadhana
saccidanandaikarase bhaktiyoge tisthati

The Lord does not change His original transcendental position, but He appears to be differently manifested according to the particular time, circumstance and society. He is understood by the linking process of Bhakti, pure devotional service.

Whatever conditions the living beings undergo in the material realm due to their imprisoned situations, the Lord also manifests His incarnations pertaining to those situations. Starting from the one cell entity known as indragopa up to the most elevated Brahma (the demigod in charge of creation), different sages have observed different types and numbers of incarnations , which are revealed in the vedic literatures.

They are seen as eight, ten, eighteen and twenty four types by different scriptures.

In the Caturveda Sikha it is stated:

vasudeva sankarsana pradyumno’ niruddhoham
matsya kurma varaho nrsmho vamano rama
ramo buddha kalkir aham iti

“The Supreme Lord Sri Krishna said: I am Vasudeva, Pradyumna, Aniruddha and Sankarsana. I am also the ten incarnations, viz., Matsya, Kurma , Varaha, Narasimha, Vamana, Parasurama , Ramachandra , Buddha,and Kalki.”

Ten different definite conditions of living beings are construed generally, from the beginning of their bondage up to the end .
Invertebrate – Matsya

Testaceous, shelly – Kurma

Vertebrate – Varaha
Crectly vertebrate, Half man half beast – Narasimha
Manikin – Vamana

Barbaric – Parasurama

Civilized – Ramachandra

Wise – Krishna

Ultra-wise – Buddha

Destrucive – Kalki
Sometimes He incarnates Himself or empowers some suitable living being to act for Him (sakty avesa avatar – empowered incarnation). In either case the purpose is one and the same. The suffering living being should go back home, back to Godhead. The happiness that the living beings are hankering for is not be found within any corner of the innumerable universes in any material planet. The eternal happiness which the living being wants is only obtainable in the kingdom of God, but the forgetful living beings under the influence of the material modes have no information of the kingdom of God.

The Lord therefore, comes to propagate the message of the kingdom of God, either personally as an incarnation or through His bona fide representative as the good son of God. Such incarnations or sons of God are not making propaganda for going back to Godhead not only within the human society, but also amongst the higher human beings called demigods as well as amongst lower species such as animal, plant , bird , insect and aquatic . This is how the vedic literatures describe the descents of the Lord, the Incarnations.

There are also svamsa (His own, unseparated) forms of Krishna, different from the forms in the spiritual realm, and these have two divisions, the Sankarshana division and the lila (pastime) avatar (incarnations) division.

From the Sankarshana division come the three purusa-avatars (Personalities who are the source and control for material creation) and from the lila avatar division come the incarnations like matsya-fish, kurma-tortoise etc.

The Satvata tantra describes that Krishna’s energies are divided into three aspects: thinking, feeling and acting. He is known differently in these three aspects:
Thinking – Krishna

Feeling – Vasudeva

Acting – Sankarshana Balarama
The material creation is due to the thinking, feeling and acting of Sankarshana Balarama.

Even though there is no creation in the spiritual realm where everything is eternal, the whole spiritual realm also depends on the acting energy of Balarama. Everything there is manifested by the Balarama or Sankarshana form.

As far as the material creation is concerned, the forms by which the energy of material nature works to bring about creation is called the Sankarshana form, and it is understood that this cosmic manifestation is created under the superintendence of the Supreme Lord.

This is further explained in the Srimad Bhagavata Purana (10.46.31):

etam hi visvasya ca bija-yoni
ramo mukundah purusah pradhanam
anviya bhutesu vilaksanasya
jnanasya cesata imau puranau

“These two Lords, Mukunda and Balarama, are each the seed and womb of the universe, the creator and His creative potency. They enter the hearts of every living beings and control their conditioned awareness. They are the primeval Supreme.”