There is a midrash that states that Pinchas and Elijah the Prophet are the same person.  What is the meaning of this?

According to the Midrash, Pinchas and Elijah the Prophet are the same person.  The simplest meaning is that the same soul descended to the world twice -- once in the body of Pinchas and once in the body of Elijah.

The same statement can be found in a number of places in Midrash.  What is interesting is that "Pinchas is Elijah" and "Elijah is Pinchas" are written interchangeably.  When Pinchas is being discussed, the Midrash says that Pinchas is Elijah.  When Elijah is being discussed, the Midrash says that Elijah is Pinchas.

Since Elijah the Prophet lived hundreds of years after Pinchas, it would apparently make more sense to say that "Elijah is Pinchas", and not the reverse.  After all, Pinchas lived before Elijah, and was Pinchas before he was Elijah.

According to an explanation in the Zohar, the soul of Elijah was actually created during the Six Days of Creation.  He has existed ever since as an angel, but on occasion, he descends to the world in human form, born of a mother and father.

This is why the Midrash sometimes uses the phrase "Pinchas is Elijah", even though Pinchas was born first.  The essence, the soul of Elijah existed before Pinchas was born.

Elijah and Pinchas led similar lives and their paths complement each other.  Pinchas is a symbol of zealousness for G-d and His commandments.  He displayed self-sacrifice to prevent G-d's name from being desecrated.  Elijah the Prophet is a guest at every circumcision, to witness the Jewish People imprint their bond with G-d in their flesh.

Through our single-minded commitment to fulfilling G-d's will, as epitomized by the deeds of Pinchas and Elijah, we will merit the ultimate Redemption, which will be heralded by Elijah the Prophet, who is Pinchas.

(Bereishis 1:20.  Yalkut Simoni, Pinchas.  Zohar, 3:15,1.  Igrot Kodesh, vol. III, p. 160.  Likutei Sichot vol II, p. 343.)