In this 2004-09-15 interview with J. K. Rowling (see also another transcript on Rowling's website), we find this question and answer.
What form does Dumbledore’s Patronus take?
Good question. Can anyone guess? You have had a clue. There was a little whisper there. It is a phoenix, which is very representative of Dumbledore for reasons that I am sure you can guess.
In what sense is a phoenix very representative of Dumbledore?
I have a couple of theories, which are personally my own, based on the extended re-reading of the HP series. I should re-iterate these are just speculation.
As we come to know in DH, Dumbledore was best friends with Grindelwald for a while in his youth. Both of them planned to control the Muggles of the world for what Dumbledore atleast believed was the "greater good" of wizardkind. However, Dumbledore saw the error of his ways and matured into one of the greatest wizards of all time and a fierce supporter of Muggle and Muggle-born rights. This is somewhat akin to a 'Phoenix transformation', in the sense that a Phoenix is always born from the ashes of its own death.
Another theory that I have nursed is fact that Dumbledore was so close and dependent on his phoenix, Fawkes. As we know, Patronuses can change in case of emotional distress. Nymphadora Tonks' Patronus changed to a werewolf in HBP when she was heartbroken after Remus Lupin turned her down. Maybe Dumbledore was pulled out of a tight spot by Fawkes, and his gratitude was so overwhelming it changed his Patronus into a Phoenix.
However, there is no evidence that states Dumbledore's Patronus was anything other than a Phoenix at anytime in the past, so my second theory could be complete horse crap.
This bit sheds some light on the issue:
"This ancient and mysterious charm conjures a magical guardian, a projection of all your most positive feelings. The Patronus Charm is difficult, and many witches and wizards are unable to produce a full, corporeal Patronus, a guardian which generally takes the shape of the animal with whom they share the deepest affinity. You may suspect, but you will never truly know what form your Patronus will take until you succeed in conjuring it."
—Miranda Goshawk's overview of the Patronus Charm
Casting of the charm:
Harry Potter: "And how do you conjure it?"
Remus Lupin: "With an incantation, which will work only if you are concentrating, with all your might, on a single, very happy memory."
Patronus then will always take animal shape (if produced full), but what it will be is unknown prior to first successful projection.
Obviously it depends on the individual and equally obviously it can transform into something else.
The only pointer one can definitely have here is the trait both Fawkes and Dumbledore share: ultimate loyalty to friends. And which said trait most probably was responsible for their bonding. It may be safely assumed that this bonding is the happy memory for AB, especially since Fawkes was a wild phoenix (which makes it next to impossible to "domesticate"it).
But there's no definite answer to that - and a slew of other related questions, as canon does not say.
A little strained, and a lot simpler than many would believe, it's the theme of self-sacrifice.
The phoenix represents self-sacrifice, rebirth, and renewal. So while there's not much to the tune of rebirth other than the continuance of his legacy of opposing evil, there's self-sacrifice.
He sacrificed himself to the cause of Voldemort's defeat. Even with him gone, hope wasn't lost, it rose again.
I don't really see why his self-sacrifice was necessary, but it fits the theme.
Dumbledore was the founder of the group: The Order of the Phoenix Lord Voldemort's first rise to power in the 1970s
Reason behind the naming the Order after the Phoenix?
Just a theory, might be wrong!
It is to signify Voldemort's evil ways of living. By renewing his life through various hideous and unspeakable ways like Unicorn murder, etc.
So, a phoenix instead, gracefully gets burnt down and gets a renewed life from it's ashes.
Thus, the name Phoenix for the Order which is established to fight against Lord Voldemort.
Dumbledore was the founder, and also (as seen in the book) has a very rare and a difficult-to-gain-trust pet, the phoenix, Fawkes.
In addition, Dumbledore is very trustworthy, just like the phoenix.
So, maybe that would be the reason behind his patronus being a phoenix.
From the evidence in the books, the most likely reason that Dumbledore has a phoenix Patronus, and that a phoenix is representative of Dumbledore, is that he clearly has a close relationship with his phoenix, Fawkes. He tells Harry that phoenixes make very faithful pets.
“He’s really very handsome most of the time: wonderful red and gold plumage. Fascinating creatures, phoenixes. They can carry immensely heavy loads, their tears have healing powers and they make highly faithful pets.”
- Harry Potter and the Chamber of Secrets, Chapter 12 (The Polyjuice Potion)
He also tells Harry that only by showing him extreme loyalty would Harry have been able to call Fawkes to his aid, further showing the connection between Fawkes and Dumbledore.
“First of all, Harry, I want to thank you,’ said Dumbledore, eyes twinkling again. ‘You must have shown me real loyalty down in the Chamber. Nothing but that could have called Fawkes to you.”
- Harry Potter and the Chamber of Secrets, Chapter 18 (Dobby’s Reward)
Fawkes helped Dumbledore in various ways, like when he needed a lookout who could warn him.
“Dumbledore was now stroking Fawkes’s plumed golden head with one finger. The phoenix awoke immediately. He stretched his beautiful head high and observed Dumbledore through bright, dark eyes.
‘We will need,’ Dumbledore said very quietly to the bird, ‘a warning.’ There was a flash of fire and the phoenix had gone.”
- Harry Potter and the Order of the Phoenix, Chapter 22 (St. Mungo’s Hospital for Magical Maladies and Injuries)
In addition, having a phoenix as a pet is very rare - it’s not like having an owl. Harry, for example, had Hedwig as his pet, but an owl wouldn’t necessarily be closely associated with him, because very many wizards had pet owls. Dumbledore’s having a pet phoenix was a more significant achievement. The note on the phoenix’s classification explains that few wizards have successfully domesticated one. Therefore, having a phoenix would be more representative of Dumbledore than the average wizard’s pet would be of them, both because of their close relationship and its rarity.
“11 The phoenix gains a XXXX rating not because it is aggressive, but because very few wizards have ever succeeded in domesticating it.”
- Fantastic Beasts and Where to Find Them
While there may or may not be some deeper out-of-universe symbolism intended by J.K. Rowling in Dumbledore having a pet phoenix, it seems likely that in-universe, his Patronus is a phoenix because of his close relationship with Fawkes.
According to "FBaWTFT" entry on Phoenixes:
The phoenix lives to an immense age as it can regenerate, bursting into flames when its body begins to fail and rising again from the ashes as a chick. The phoenix is a gentle creature that has never been known to kill and eats only herbs. Like the Diricawl, it can disappear and reappear at will. Phoenix song is magical: it is reputed to increase the courage of the pure of heart and to strike fear into the hearts of the impure. Phoenix tears have powerful healing properties.
Many of these listed properties can be claimed to be representative of Dumbledore:
Old wise mage
Survived being killed by Snape err... sorry too many fanfics read.
Gentle person, shown in canon as very kind and soft spoken despite great power
Never kills (even Grindewald, or Voldemort, or Death Eaters - which is arguably dumb and caused untold damage but that's beside the point).
Increases the courage of his followers and is a leader of the Light side (pure of heart) and is feared by the Dark Side.
According to "Fantastic Beasts: The Crimes of Grindewald" script, Dumbledore family and thus he himself had affinity with Phoenixes:
Well, I’ve always felt an affinity with the great magical birds. There’s a story in my family that a phoenix will come to any Dumbledore who is in desperate need. They say my great-great-grandfather had one, but that it took flight when he died, never to return.
(Scene 28)
My own interpretation of why Dumbledore is represented by a phoenix is a bit more complicated, and a touch darker. Phoenixes do represent self sacrifice, as other answers mentioned, and loyalty makes sense looking at Dumbledore and Fawkes, and perhaps Harry (chosen of a Fawkes-feather wand).
I do like Fawkes, his character and actions, but I think there is a reason that Tom Riddle was also chosen by a phoenix feather wand and carried it faithfully and without issue through his first rise and attempted conquest of the wizarding world. Phoenixes are creatures of fire - and fire is not always gentle. I don't recall anything in canon, abut which traits qualify him, but I would guess the literal connection with death as he was killed and reborn several times from the ashes; a loyalty - to a cause or idea in his case, I think his cause was eventually lost beneath soul damage insanity but he probably had a good one if he managed to recruit so many to begin with; self sacrifice, in that he's willing to lop of bits of his soul (which is not useful, or generous, but a sacrifice of the self nonetheless); a kind of vision of what's best for the wizarding world, and a willingness to sacrifice to make it true.
To bring it back to the original question, these are traits I see in Dumbledore as well. He is willing to use self-sacrifice in his goals, but also willing to sacrifice others - notably Harry himself, and generally leaving students in danger throughout the books because it fits a plan. The more fanatic version of his 'greater good' philosophy he held when he was running around with Grindlewald, would fit a similar profile to a young Tom Riddle - I would not be surprised if he had a phoenix feather wand originally. I see the fire symbolism strongly in that kind of fanaticism, as something that could achieve great things, but could also be intolerant and cruel. Fawkes as an individual might not agree with that kind of mindset, which would explain why he came to Dumbledore only after he split with Grindlewald and tried to rework his philosophy - but the wand in Riddle's hand shows it's not incompatible with his base nature.
I think Dumbledore is represented by the phoenix because he fits the symbolism in a complex way. Riddle gets a lot of the darker fire symbolism with a heavy emphasis on fanatic and sacrificing others, Harry gets a lot of the lighter symbolism self-sacrifice, and loyalty to a cause (that doesn't turn to sacrificing others). There's enough of the fire nature in them both to make the phoenix feather wands understandable. Perhaps Dumbledore qualifies to be represented by a phoenix instead of just a feather because he draws from the different qualities unevenly, or because he would like to choose which qualities to honor, as Fawkes seems to. Or perhaps it is only because he believes so strongly, past the edge of his own death.
