Reincarnation is really a Hindu concept, though the Buddha used it to explain his teaching to Hindus. Reality, from a Buddhist standpoint, is somewhat different, both from the Hindu religious concept, and from the modern concept of death and birth.
It's really hard to understand, and I've tried explaining it to several people before, to no avail - if you haven't trained your mind in just being mindful, it is difficult to see. But, anyway, here's my small change on reincarnation:
Close you eyes and forget all of your beliefs and ideas, forget science and culture and what your parents taught you. You will come to see after a short while that there is, scientifically, only a maximum of six different kinds of phenomena - seeing, hearing, smelling, tasting, feeling, and thinking. This realization is made much easier by fixing the nature of the phenomena firmly in your mind simply as "seeing", etc., according to Buddhist meditation practice. It is clear from this exercise that so many of our ideas about space, time, and reality are just concepts or, at best, extrapolations of reality. Take time, for instance. In reality, there is only one moment - neither the past nor the future exists outside of this one moment. And this moment is eternal - whether we die, or are born, these are just concepts, like the word "wave" is just a concept used to describe a movement of part of the ocean. But no matter how often the waves crash against the shore, it is still the same ocean. Death is only the crash against the shore - really, nothing has died, it is a physical process which is simply the dissolution of a collective structure of matter. In reality, there are still only six phenomena, just as the wave is still only water. And just as the waves come again and again, so to does the process we call "death" come to beings again and again. At that moment, the clinging mind simply continues its search for happiness among the six senses, and follows after new experiences as they arise, just as it always has. The experiences change, of course, but they are still the same six phenomena. The functioning of the six senses in the present moment never ceases for one who still seeks after more pleasureable phenomena. So, really it is not that Buddhists believe in rebirth, it is that they don't believe in death :) The attainment of Enlightenment simply means realizing that this world is made up entirely of an endless process of arising and ceasing phenomena that is entirely unsatisfying, and thereby not falling into suffering when the beloved disappears or the unbeloved appears. This realization also leads to final freedom from the process of arising and ceasing phenomena, as no more craving means no new fuel to create new seeking out and new phenomena. Or, in the case of a mind that merely craves less, less fuel, less phenomena and a simple life, free from gross forms of suffering.
There. That was two cents worth, surely.