What is a "phoenix" and how is it used in the Fahrenheit 451?

In Greek mythology, the phoenix is a large bird with the ability to be reborn. To do this, the phoenix builds a flammable nest called a pyre and allows its body to be consumed by the flames. After three days in the fire, the newly-reborn phoenix emerges from the ashes of the flames.
In Fahrenheit 451, this mythical bird is used by Bradbury as a symbol. Firstly, it is used as a symbol of censorship. You will notice that Montag (and the other firemen) wear the image of a phoenix on the sleeve of their shirts. In this context, the phoenix represents the rebirth of this society, after censorship and book-burning were introduced.
In addition, the image of the phoenix acts as an allusion to the fire, which is used by Montag and his colleagues to burn illegal books.
At the end of the novel, the phoenix takes on a new symbolic meaning, as explained by Granger:

There was a silly damn bird called a phoenix back before Christ, every few hundred years he built a pyre and burnt himself up. He must have been first cousin to Man. But every time he burnt himself up he sprang out of the ashes, he got himself born all over again. And it looks like we're doing the same thing the phoenix never had. We know the damn silly thing we just did.

To put this quote into context, Granger and the other book covers are watching the city being destroyed by firebombs. This makes Granger think of the phoenix because, although the city is being destroyed, its death will provide the opportunity for a rebirth, just like the mythical bird. It is Granger's hope that, instead of creating another society filled with censorship, he and his men can create a new society which is based on learning, books, and independent thought. So, instead of repeating history, like the phoenix does, Granger hopes that society can learn from its past mistakes.
The symbolic meaning of the phoenix, therefore, changes as the novel progresses. While it begins as a symbol of the fire and censorship, it takes on a new meaning at the end of the novel because there is hope that the city can be reborn, just like the phoenix. Unlike the phoenix, however, there is a realization that society must learn from its past mistakes and never let censorship take over again.