The Hindu concept of reincarnation is complex. To really understand it you cannot discuss it in a vacuum. You have to put it into context with the other concepts that go along with it.
To understand the Hindu concept of reincarnation we have to start with the relationship between the Atman, our "soul" and Brahman, ultimate reality. Despite that fact we perceive these things as separate they are in fact one and the same, the concept of the individuated self, "I", is an illusion. Hindus believe that we start out ignorant of reality.
What makes us ignorant of reality? Maya is the illusion that tricks us into believing that we are separate from the rest of the universe, that Atman and Brahman are different, unrelated things. Maya shrouds our "souls" in layers of illation that separate us from the rest of reality. .
Working in conjunction with Maya is Samsara. Basically Samsara is the cycle of reincarnation our "souls" are trapped in. Hindus believe that when we die our "soul" moves on to another body. It should be mentioned that Atman or the "soul" is not the western idea of the soul as being our personality, Atman is, as you recall, one with the ultimate reality and Hindus belief that our personality is just another layer of the illusion that holds us in Samsara and that it does not survive past death. So our personality is not sent on to the next life, but there is continuity and something infinitely subtler is sent on.
Karma is what determines how our "soul" reincarnates. Karma is created by everything we do, think or desire. These things can create good or bad karma. Good karma means we reincarnate to a higher life form, bad karma means we reincarnate as a lower life form. Essentially though Karma weighs down our "soul". With Karma less is more, karma is created as long as we are trapped in Maya so good karma means less karma. The idea of Samskaras goes along with Karma; it is the mechanism by which Karma is carried out.