It is widely accepted among non-Christians, probably because it appeals to many who would like to believe that they would be given a second chance in case they failed to make the grade in this life. Christianity disputes reincarnation because it is unnecessary, since anybody can "make the grade" simply through an act of their own will through faith in Jesus Christ and good works (James2:17). The scientific rebuttal to reincarnation is quite simple. Because of the population explosion, more people are currently living on the earth than have ever lived on the earth for the entire history of humankind. In other words, over half of the people who have ever lived on earth have never died even once! There simply are not enough dead souls to go around for a second time. This does not absolutely eliminate reincarnation, but it does severely restrict its extent, especially for those who have claimed to have lived several times before. However, some people believe that souls can be reincarnated into or from animals. In that case, it is possible that many people have been frogs before they became princes.
The one passage that some point to as evidence for reincarnation is Matthew 17:10-12 which links John the Baptist with Elijah. However, the passage does not say that John the Baptist was Elijah reincarnated, but rather that he would have fulfilled the prophesy of Elijah coming if the people had believed his words and thereby believed in Jesus as the Messiah (Matthew 17:12). The people specifically asked John the Baptist if he was Elijah, and he said "No, I am not" (John 1:21).
Conclusion: Reincarnation is a popular belief among many "spiritual" people, and is commonly held in most major religions. It has such popularity because people would like to believe that they will be given a second chance if they "blow it" in their first life. In Judaism, where salvation is based upon "being good," one could be condemned quite easily by making some major mistakes in his life. The "hope" of reincarnation provides an escape from a God who demands righteousness. However, both the Old and New Testaments do not leave reincarnation as an option that God chose to use. Why would God not allow a second chance for those who made mistakes on their first attempt? The answer is quite simple. Salvation is a free gift for all who want it. It requires only repentance from your former life (admitting you were wrong and wish to change) and belief in the completed work of Jesus Christ on the cross to atone for your sins. For the Christian, however, there can be no doubt: reincarnation is unbiblical and must be rejected as false. Anyone can be saved through the gospel (good news) of Jesus Christ. Don't expect a second chance to go to heaven. Today is the day of salvation!