Dear Friends, I have very little knowledge in Sanskrit, but i can expalin this to you all. Jai SitaRam means Sita RamChandra swamy Victory. There is Inner meaning in it. "Jai"means Victory, Sita means Bhudevi's Child, Rama means "Ra" means pleasure, and Ma means Manas, or dil or brain.Bhudevi means earth you know that very well, right? So, Complete meaning is "The Victory of everyone's peace of mind on the earth". Here, you can ask how come the meaning comes to this? Sita means child of earth, like anyone of us. Rama means mental pleasure or mental peace. Jai means Victory. So, are you agree with me?
In villages like in India people use to say Jai SitaRam, when one meets any other person.
They use to think "Sarvejna Sukhino bhavanthu" = All should be happy.
Jai SitaRam or Jai SiyaRam or Jai Ramjiki.