Karma and reincarnation are related beliefs that have been popular in India and surrounding countries for a very long time.
Karma means action. The idea is, at it's simplest, that good is rewarded and evil punished. Or more practically put: selfishness is punished and unselfishness rewarded. The Dalai Lama puts it even more radically: From the perspective of karma it is selfish to help others, because that will be rewarded.
Belief in karma means that reincarnation follows logically: otherwise the actions done just before death would not have their logical result.