What is Reincarnation?
The soul is the living matter, while the body is the physical being which houses our soul. The proponents of the concept of reincarnation say that the body is mortal while the soul is immortal. So, when any living being dies, it is only the body which dies. The soul merely leaves the body and assumes a different life form.
Each culture has its own, slightly different interpretation of it. Some people believe that a soul which was contained in a human body will always be contained in its next earthly life. Other cultures believe that a soul goes through the cycle where it lives in the bodies of all the life forms on Earth.
While there are certain verses in the Holy Bible which can be interpreted to be hinting vaguely at it, such as, John 3:3 - No one can see the kingdom of God unless he is born again, and there is also the part where John the Baptist is supposed to be the reincarnation of Elijah, the Bible doesn't make any clear reference to the reincarnation of a soul.
These are simply the things which merely hint at the idea, but don't clearly state it. The Holy Bible merely says that after a person dies, it is the body which dies, while the soul goes into Heaven for the Last Judgment. The Last Judgment is done in all fairness, and depending upon the life that the soul has led, it is sent to Heaven, Hell, or Purgatory.
Most people who follow Christianity, think of resurrection and reincarnation as one. However, they are different. Reincarnation is the change in the body which houses the soul after death.
Resurrection, on the other hand, is a case wherein the soul unites with the body it contained - the body which just died recently - temporarily for the sake of the Last Judgment. When the Last Judgment is made, both the body and the soul unite for one last time to receive the Judgment.
Doubters and protagonists of the reincarnation theory are still finding verses in the Bible that vaguely hint at it as a real possibility. The verses they are talking about are merely open to interpretation, and do not say anything which assertively endorses it.