"About time, too. He's been looking dreadful for days; I've been telling him to get a move on."
-- Albus Dumbledore about his phoenix Fawkes, after Fawkes burst into flames (CS12)
Fawkes was the pet phoenix of Albus Dumbledore, who used what appears to be one of Fawkes’ scarlet body-feathers as a quill (OP38). The wands of Harry and Voldemort each contain a feather from Fawkes’ tail (GF36). When Fawkes is asked to keep watch and give warning, or deliver messages, he leaves a single golden tail feather, which Dumbledore, for one, is careful not to leave lying around (OP22).
Fawkes' age is unknown, but Dumbledore was his first owner (ITV), and he must have been around by 1938 since that is when Tom Riddle acquired a wand containing one of Fawkes' tail feathers.
Owned by Albus Dumbledore.
A phoenix can lift a great weight with its tail in flight; when Harry grasped Fawkes' tail in the Chamber of Secrets, he felt as if he were becoming weightless - which is saying something, as Ron, Ginny, and Lockhart were in turn hanging on to him(CS12, CS17).
Ability to disappear and reappear at wil in a burst of flames (FB, CS17). Fawkes has been known to deliver messages in this manner so quickly that only a flash of fire in the air has been seen (OP22).
Phoenix tears have healing properties (FB, CS12). Fawkes twice used his tears to save Harry after he had been bitten by a venomous magical creature (CS17, GF36).
The song of the phoenix gives strength and hope to those for whom it sings, 'increasing the courage of the pure of heart and striking fear into the hearts of the impure' according to Scamander (FB). When wands sharing a phoenix-feather cores from the same bird are forced to do battle, the rare Priori Incantatem effect that results will be accompanied by the sound of phoenix song (GF34).
The phoenix's most unusual characteristic is that after its body begins to fail, it dies in a burst of flame, to be reborn from the ashes. This occurs not only on a phoenix's natural Burning Day, but also if the phoenix receives a fatal injury, such as being hit with a Killing Curse (OP37).
Other canon notes and references
As a phoenix approaches its Burning Day, it resembles a half-plucked turkey and makes gagging noises. Its eyes become very dull and its feathers fall out. Then suddenly it will burst into flames and turn to a pile of ashes on the floor. From this pile of ashes rises a newborn phoenix, small and ugly. In a matter of days it becomes a fully grown, beautiful phoenix.
On 5th November every year, Britain celebrated Bonfire Night (sometimes called Guy Fawkes Night) the Gunpowder Plot, in which Guy (Guido) Fawkes and other extremist Catholics plotted (but failed) to blow up James I and his Parliament. People have firework parties or attend organized displays, and effigies of Guy Fawkes (known as "the guy") are burned on bonfires. The term bonfire is derived from 'bone fire' because, originally, bones were the primary material burnt.
Fawkes (and phoenixes in general) has brilliant crimson and gold coloring. Interestingly, these are also the colors of Gryffindor House. Further, phoenixes and Gryffindor are both associated with fire. -BB
The only two wands containing feathers from Fawkes belonged to Voldemort and Harry Potter (GF36). This is one of several commonalities between the two adversaries. -BB
Phoenixes are immortal. Voldemort seeks immortality, and a major theme of the series is coming to terms with the reality of death. Fawkes is the only character who can exist forever, but in doing this he must repeatedly suffer death. -BB
In the Iliad and other texts about the Trojan War, Phoenix was the name of the man who was in charge of raising and educating the hero Achilles (source: Wikipedia). Dubledore's role in Harry's life is underscored by the former's close association with a phoenix. -BB
Fawkes has been through this regeneration as the result of being hit with a Killing Curse intended for Dumbledore (OP37), so the effect is not limited to his normal Burning Days.
Curiously, Fawkes was not affected by the gaze of the basilisk in the Chamber of Secrets; given the rarity of both species, it is possible that no such confrontation had ever before occurred, so it isn't surprising that Scamander does not record the phoenix's immunity.
Given that the gentle nature of phoenixes is well known (FB), it's noteworthy that Fawkes entered into direct combat with the basilisk at all, as phoenixes are not predators and have little need for fighting skills even in their own defense.