Meaning
Jaya means victory or glory, Shri is an honorific of auspiciousness, and Ram names the prince of Ayodhya whose story Valmiki tells in the Ramayana. Together the words acclaim: glory to the noble Ram. Vaishnava devotion hears in them less a shout of conquest than a household greeting, an affirmation that dharma prevails and that the Lord's name is enough.
जय श्री राम।