1784, "descent of a Hindu deity," from Sanskrit avatarana "descent" (of a deity to the earth in incarnate form), from ava- "down" + base of tarati "(he) crosses over," from PIE root *tere- "to cross over" (see through). In computer use, it seems to trace to the novel "Snowcrash" (1992) by Neal Stephenson.
In Hinduism, a god made visibly present, especially in a human form. The Buddha is considered an avatar of the god Vishnu.
Note: By extension, an “avatar” is any new embodiment of an old idea.