Reincarnation is as its most basic, the belief that at the very least some part, and in some cases the entire, soul and/or sprit and/or mind is carried on after death to be reborn into another life. At this point various opinions make nothing completely agreed upon. The species that a soul may be reborn into to, the time that it takes to be reborn, weather past lives affect future lives (usually in terms of how high up in the social or species ranking you are born into), memories one may take with you, whether lessons are to be learned in each life, if the cycle ever started or will stop, if new lives are ever created, if the system is random or ordered. The list goes on and on.
This belief is most commonly held among those of eastern religions such as Buddhism, but people can believe in reincarnation in practically any spiritual system that does not clash with their afterlife believe.
Reincarnation is often used to explain therianthropy, people either claiming to have been their animalside's specie in a past life (or lives), or that something got a bit messy in transition, so that all or part of an animal mind got stuck in a human body. The fact that energy can not be created or destroyed is also used in explanation for reincarnation, though scientifically, this doesn't explain it.