Chamber of Secrets is the first Harry Potter book in which we meet Fawkes, the phoenix. Fawkes lives in Professor Dumbledore's office. Harry sees Fawkes on a Burning Day. He starts out looking old and sick. Suddenly, he bursts into flames and burns down into a pile of ash. Harry is utterly shocked and immediately apologizes to Professor Dumbledore: "Your bird – I couldn't do anything – he just caught fire —" (12.14). Professor Dumbledore explains: "Phoenixes burst into flame when it is time for them to die and are reborn from the ashes" (12.18). Professor Dumbledore also tells Harry that phoenixes can carry immensely heavy loads and that their tears have healing powers. Above all, phoenixes are loyal birds. They stand by their chosen people.
Phoenixes generally represent rebirth (for obvious reasons). In addition to endurance, this particular phoenix symbolizes faith. When Harry declares his honest, uncomplicated loyalty to Professor Dumbledore in the Chamber of Secrets, Fawkes appears to help Harry survive the basilisk poison.
As for the link between Fawkes and Professor Dumbledore, Professor Dumbledore is also loyal (consider his protection of Hagrid, for example). He does his best to help Harry carry the metaphorical burden of his persecution by Voldemort – OK, maybe that's not a literal heavy load, but we think it works in the language of symbolism. Professor Dumbledore may not heal Harry physically, the way Fawkes does, but he often helps Harry feel better. So, Professor Dumbledore shares some of the qualities of his phoenix. J.K. Rowling underlined these similarities by announcing that Professor Dumbledore's Patronus charm takes the form of a phoenix, "which is very representative of Dumbledore for reasons that I am sure you can guess" (source). For more on phoenix symbolism, check out our Learning Guide to Harry Potter and the Order of the Phoenix.