As Aaron's answer makes clear, there is no mention of John the Baptist in the Tanach, nor in Jewish tradition, and especially no mention of him being a reincarnation of Elijah the Prophet. He is a figure of the Christian Bible.
There is, however, an old Jewish tradition that Elijah will come to prepare the way for the Messiah. This is mentioned in Malachi 3:23 ("Behold, I send unto you Elijah the Prophet") and in the Talmud as well (Eiruvin 43b).
Most likely, the writers of the Christian Bible knew of this tradition and therefore sought for a way to make it work with the life of Jesus, in attempts to "prove further" his status as Messiah. Since Elijah the Prophet did not come before Jesus to announce him as the Messiah, they had to say that John the Baptist did this job as a reincarnation of Elijah.
Interestingly enough, in John 1:21 in the Christian Bible, John the Baptist replies "no" to the question of whether he is Elijah the Prophet or not. So there is a contradiction between this and Matthew 11:13-14 and 17:12, where Jesus says on two occasions that John was Elijah.
On a side note, Rambam makes it clear that Elijah coming before the Messiah is not necessarily a requirement of the Messianic Age (Hilchot Melachim 12:2), although most Sages maintain it is. Despite what Rambam says, though, the idea that Elijah's coming will proceed the coming of the Messiah is a widespread tradition in Jewish circles.