Buddhists don't believe in reincarnation, but in rebirth -- the are rather different concepts. Hindus believe you'll physically become some new person, animal, plant or inanimate object (that your soul is "transmigrated" into a new body); Buddhists believe that only your "karma" survives death.

In some versions of reincarnation, you get some freedom of choice about your next life, by choosing, in between "lives", the body / circumstances into which your soul will be "installed". Buddhist rebirth gives you no choice; everything is determined by your "karmic tendencies" and the person receiving those is not "you" in the sense we typically understand the self; it is considered an entirely new being, not a retread of an old one -- albeit, burdened or blessed with past karma.

These distinctions get blurred by different sects in Hinduism and Buddhism, and by certain exceptions. For example, the leader of Tibetan Buddhism, the Dali Lama, is, to my understanding, literally reincarnated, but this is considered a special case.

Many indigenous people believe in it, as well as Asian religions, and sects within Christianity and Islam. It is also mentioned several times in the Bible. Jesus even said John the Baptist was a reincarnation of the prophet Elijah. Hebrews, in the New Testament, also says Jesus appeared in the Old Testament in another reincarnation; the priest Melchizedek.

[en.wikipedia.org]