Originally, jayaśrī (in IAST script) is a Sanskrit word. Jayashree is a popular Romanization for it. Many modern Indian languages may use it - Bengali surely does in the literary register. Probably Hindi does too. However, the pronounciation in the respective languages would be adapted to their phonology more or less. In Bengali, the pronunciation is something like "jɔyosri" (the two "a"-s are pronounced differently). In Hindi, I suspect, it would be something like "jaiśrī" (i.e. the second "a" would be deleted). It is also a common given female name in the corresponding cultures.
jaya means "victory".
śrī has two main, probably related, meanings: 1) another name for the goddess Lakṣmī, 2) beauty, splendour, prosperity (in Bengali usage "beauty" is the dominant sense).
So, it can either mean simply the "splendour of victory" or Lakṣmī in her aspect of the bringer of victory. In Bengali usage, my feeling is that it is used rather as an elevated synonym for "victory", maybe mildly deified sometimes.
