Essentially, reincarnation is the belief that death is simply a doorway and we will be born into another lifetime. In my personal beliefs, I hold this to be true. There have been some fascinating tales told of those who have been aware of this cycle, both from our culture and those who embrace reincarnation.
Most people simply assume that they have past lives. This isn't necessarily true. I've known two people that are on their first lifetime. One is aware of this fact, the other is not. And the one who is aware is certain that this is her only incarnation into this reality.
We aren't part of a cosmic treadmill where we die, are born again, only to start the process all over again. Some of us are like the birch tree. We are born in the springtime, mature in the summer, and die in the winter of our lives. When our personal springtime comes again, we are born once more, building on the lessons of previously lifetimes. However, some of us are like flowers that bloom once and are never born again. The deciding factors are the desires of your soul and how being physical will influence your spiritual growth.