Tibetan sky burial, or jhator, is the practice of leaving the bodies of the dead for the birds to eat. The body is usually chopped into pieces to make things go faster.
This practice probably began because, while in Tibetan Buddhism (and Buddhism in general) cremation is the preferred method of disposing of the dead, there's just not enough wood on the Tibetan plateau for every Tom, Dick and Harry to be burned. Only the rich people can afford such luxury. While burial would also be an acceptable option, the hard rocky ground and a layer of permafrost makes this difficult.
Sky burial is not uncommon in Asia, and Buddhist sky burial probably originated in Tibet. The practice is first described in the Tibetan Book of the Dead (Bardo Thodol). The exact origins of this text are a little cloudy, but it may have been put down as early as 800 CE. China tried to outlaw the practice in the 1960s, but by the 1980s it was legalized again. In Middle Eastern and Central Asian countries where Zoroastrianism is (or was once) practiced, the cultural reasons for sky burial are different, and are probably unrelated.
One other form of body disposal in Tibet is mummification. This is used for the highest religious leaders, like the Dalai Lama.