Phoenixes are magical birds that age, die, and are reborn from their own ashes. They symbolize endurance and new beginnings. Professor Dumbledore has an actual phoenix, Fawkes, who swallows an Avada Kedavra meant for Harry, dies, and returns as a "tiny, ugly, featherless" (37.20) chick at the end of Book 5. Fawkes the phoenix has saved Harry's life before: his tears heal Harry from the basilisk bite in Book 2, and he helps to soothe Harry's shock after the confrontation with Voldemort and the death of Cedric Diggory in Book 4. As such, Fawkes is a recurring character in the novel; he is endlessly loyal to both Harry and Professor Dumbledore, and he seems generally aligned to the cause of good.
Fawkes also seems symbolically connected to Harry's emotional rebirth at the end of Book 5. Indeed, Harry is something of a phoenix, given that he was found living in the ruins of his parents' old house when Voldemort tried to kill him in the first place. He has also nearly died and risen again several times: after his showdown with Professor Quirrell, with the basilisk, and with Voldemort himself at the end of Book 4. In this case, however, Harry's rebirth is more metaphorical. When Professor Dumbledore meets with Harry in the second-to-last chapter of the novel, Harry finally vents all of his frustration at the situation in which he's been placed. It's like the lancing of a wound, and all of the dark resentment and suspicion against Professor Dumbledore that he has been bottling up comes spilling out.
By the end of the meeting, Harry has come to some kind of peace that soothes the violence of his grief and bitterness. He begins to return to the Harry Potter we know and love: older and wiser, perhaps, but less gloomy and angry than he has been throughout Book 5. Accompanying Harry's own emotional rebirth is Fawkes's physical rebirth: Fawkes took the Killing Curse meant for Harry and returns to a venerable, ugly chick. Harry has been broken down emotionally and is taking the first steps to learning to trust Professor Dumbledore and his friends again.