To be reincarnated literally means to be put into another body after you leave the one you have now. In practical terms people use everyday, it is treated as a mystical force that rewards good and punishes evil. In more philosophical/religious terms, when one person harms another, the victim is believed to deserve it based on a past life, while the perpetrator is earning bad karma and will be the victim next time around. So in theory this cycle can never end, since the last person to pay off karma will only do so by means of someone else earning bad karma.
Karma cannot be proved; it cannot be measured; its mechanism cannot be explained. Thus it is a blind faith, and one I find at least as irrational as any theistic faith. Who manages karma? Who weighs people’s lives? Who decides what they come back as? How does this return happen? How can anyone believe karma can be compared to some mindless physical process or chemical reaction, both of which happen to be highly complex and evidence of intelligent design? To think this all happens by unguided processes is simply a wish or magic. And why is God bad for not preventing evil, yet karma is good because evil people are being repaid for what they can’t remember doing in a past life?
The teachings of the Bible leave no room for reincarnation. Reincarnation is, simply put, salvation by works. I view it as an endless, hopeless, purposeless treadmill. In contrast to reincarnation, I find great hope and purpose in knowing that I only have to go through this blood-stained life once and then live in eternal bliss.
The Bible makes it abundantly clear that none of us can ever be good enough by our own determination, so for it to also teach reincarnation would be to consign us to a living hell of straining for a goal we can never reach. Salvation by works and salvation by faith (Eph. 2:8-9) are hopelessly incompatible. Why would I want to trade a priceless gift for a wage that will never be enough to buy eternal happiness?
Jesus came in the flesh to die once physically, then got his new immortal body at the resurrection. He never had any other bodies and will never need any more. In the same way, the Bible says we who follow Jesus will also trade our mortal body for an eternal immortal one.