Reincarnation is a Hindu concept and rebirth is a Buddhist concept.
Although there is really no one religion that is “Hinduism,” because it has so many different ways of manifesting itself, a central teaching common to the religions commonly referred to as Hinduism is that the Self is re-born time and time again until it merges with Brahma.
The Buddha did not teach the doctrine of reincarnation. It teaches that people are separate from Brahma and that they must strive towards Brahma in order to merge with It some day.
The Buddha’s teaching is much simpler. Nothing could be simpler than what the Buddha taught. He simply taught that there are no two things. No independent Self, no Brahma. If there were a Self and a Brahma, then there would be two things.
Rebirth occurs when conditions are ripe for the five aggregates that make up a sentient being to come together, again creating the illusion of an independent self that in reality is merely a collection of form, feeling, thought, choice, and consciousness.
Thus, no eternal Self reappears because there is no eternal Self.
The self that is empty of independent existence is in constant flux during all of its apparent lifetimes. The five aggregates that reappear are also in flux; there is no eternal Self that is reincarnated but there is an apparent self that is reborn. One deluded thought, one deluded lifetime, leads to further deluded thoughts and further deluded lifetimes.
The illusion of an independent self continues to reappear when conditions are right, just as hurricanes are re-born when conditions are right. One becomes a Stream Winner when the emptiness of Self is realized.