First Meaning: -
Srila Jiva Gosvami reveals the meaning of each of the sixteen Names in the Hare Krishna Mahaamantra in his Mahaa-mantrartha Dipika as follows: -
hare Krishna hare Krishna Krishna Krishna hare hare
hare Raama hare Raama Raama Raama hare hare
(1-Hare)
Krishna steals the minds of everyone, yet Radha steals even His mind by Her divine expertise. Thus She is known as Haraa.
(2-Krishna)
He forcibly attracts Sri Radha with the sweet sound of His flute, therefore that Lord of all enchanting qualities is known as Krishna.
(3-Hare)
It has been heard that during the rasa-lila, doe-eyed Radha was stolen away by Krishna to be alone with Him in a secret forest bower. She is therefore known as Haraa.
(4-Krishna)
When Krishna sports with Radha, Her golden hue takes on the dark complexion of Krishna's skin. He is thus known as Krishna.
(5-Krishna)
In order to please Sri Radha, Krishna manifested the most wonderful lake (Syama-kunda) in Vrndavana. He then called all the holy rivers to fill it. He is thus known as Krishna.
(6-Krishna)
By Her unsurpassed love, Radha charms He who performs wonderful lilas on the banks of the Yamuna. Therefore, those who are sober know Him as Krishna.
(7-Hare)
While in Gokula, Sri Hari (Krishna) killed the demon known as Aristasura. During that time, Radha cried out to Him with great feeling and by doing so, She stole His mind. She is thus known as Haraa.
(8-Hare)
Filled with ecstatic love, Radha sometimes sings the glories of Hari’s exploits quietly and sometimes She sings them aloud. Those who are expert in the secrets of divine sentiments call Her Haraa.
(9-Hare)
Due to the intense love of Sri Radha, Sri Hari becomes so captivated that His flute falls from His hand. With the desire to enjoy in the forest bowers with Krishna, Radha steals His flute. That goddess is thus famous as Haraa.
(10-Raama)
Krishna, who is expert at embracing, sports with Radha in the forest groves or in the caves of Govardhana. Thus He is known as Raama.
(11-Hare)
That most merciful Radha destroys the miseries of Her devotees and gives them great happiness every day. Therefore that goddess is known as Haraa.
(12-Raama)
The minds of the devotees are continuously drowned in an ocean of supreme joy by seeing the beautiful dark form of Krishna. Therefore He is known by the Name Raama.
(13-Raama)
Radharani is known as Raama because She enjoys loving pastimes with Acyuta (Krishna) in a secret forest pavilion. Since He is always by Her side, He is known as Raama.
(14-Raama)
When the residents of Gokula were crying due to fear of the forest fire, Krishna immediately swallowed it and gave His devotees great joy. In this way, He is known as Raama.
(15-Hare)
Sri Krishna went to Mathurapuri in order to destroy the demons. However, due to being captivated by the love of Radha, He later returned. Therefore She is known as Haraa.
(16-Hare)
When the son of Maharaja Nanda returned to Vraja, He took away the suffering of all the Vrajavasis. By His wonderful exploits, He steals the heart of Sri Radha. Thus He is known as Hari.
Second Meaning: -
There are three Holy Names that make up the Maha-Mantra, Hare, Krishna, and Rama.
1. 'Hare' is a powerful word that directly addresses the energy of the Lord
2. 'Krishna' is the Name of the Supreme Personality of Godhead. Krishna is the Possessor of every quality in and out of the universe. He also possesses these qualities in an unlimited quantity.
3. 'Rama' means the reservoir of pleasure.
By chanting this Maha-Mantra and doing service to Krishna, anyone can be linked to this reservoir and get unlimited happiness.
While engaged in nama-sankirtana one should chant with deep feeling, full faith, firm determination, and the realization that Radha-Syamasundara are personally present in the maha-manlra with all of their transcendental beauty, qualities, and pastimes. Such performance of nama-sankirtana will surely bring the desired result of Krishna-prema