Fawkes seems to be a very special character.
First, he is the one who gives the feathers that will later be used in both Harry's and Voldemort's wands.
He plays a special role in Harry Potter and the Chamber of Secrets: he's the bird that helps Harry get rid of the basilisk, the monstrous snake that lives in Hogwarts and is controlled by Tom Marvollo Riddle's diary through Ginny Weasley. Fawkes blinds the basilisk and, with its tears, dresses the wound made in Harry's shoulder. Then, he transports Harry, Ginny, Ron Weasley (her brother) and Gilderoy Lockhart, their Defense Against the Drak Arts teacher.
In the fifth book, Harry Potter and the Order of the Phoenix, Fawkes is a special part of the Order, a phoenix himself. At Dumbledore's office, in the night Harry sees Mr. Weasley being beaten by a snake (Voldemort's snake, Nagini), Dumbledore asks Fawkes to advise him if Umbridge goes to the office.
Furthermore, his colours are red and gold, and these are Gryffindor house's colours. Some fans believe that this could not be a coincidence as the author, J. K. Rowling, doesn't use similarities without specific meaning.