"It is widely believed in wizarding circles that Sir Cadogan was one of the famous Knights of the Round Table, albeit a little-known one, and that he achieved this position through his friendship with Merlin. He has certainly been excised from all Muggle volumes of King Arthur’s story, but wizarding versions of the tales include Sir Cadogan alongside Sir Lancelot, Sir Bedivere and Sir Percivale."
-- Pottermore "Sir Cadogan" (Pm)
King Arthur was a legendary leader of Great Britain, ruling from his Round Table of Knights at Camelot and Avalon. His queen was Guinevere.
Arthur was probably a Muggle, but he did have a sister named Morgan Le Fay or Morgana who was a Dark sorceress and a bird Animagus (FW). However, she fought with Arthur’s mentor and teacher, the great wizard Merlin (FW), whose name lives on in the Order of Merlin, an award given since the 15th century: First Class for bravery, Second Class for achievement, and Third Class for knowledge or entertainment (Pm: Order of Merlin).
A friend of Merlin's was Sir Cadogan, seen in a painting that protected the Gryffindor Common Room for a while in Harry's third year (PA6, PA9). He was one of Arthur's Knights of the Round Table, along with Sir Lancelot, Sir Bedivere and Sir Percivale. He was a good man in a fight but "hot-headed and peppery" (Pm) He is not included in Muggle versions of the King Arthur tales, but is remembered by the Wizarding World for his fight against the Wyvern of Wye, a fierce type of dragon, whom he charged on his fat gray pony and blew up with a broken wand. (Pm: Sir Cadogan).
Brought unity to England.
Arthur from Medieval Latin Arthurus/Arturus, from Welsh arth "bear," cognate with Greek arktos, Latin ursus
Note on dates: Some histories list King Arthur living in 500 or 600 AD. The Famous Wizard Card for his mentor Merlin says "Medieval" but lists him as a Slytherin, and that could only be possible after Hogwarts was founded in 990 AD. We can also cite this from Wikipedia concerning Merlin: "The standard depiction of the character first appears in Geoffrey of Monmouth's Historia Regum Britanniae, written c. 1136." In light of all that, we are assuming that King Arthur and his contemporaries in the canon, such as Morgana and Sir Cadogan, also lived circa 1000 AD.
Several motifs from the Arthurian legends are found in the Harry Potter books:
Arthur Weasley shares a name with the famous king, and his daughter Ginny or "Ginevra" seems to be named for Queen Guinevere. Her brother Percy may be named for Sir Percivale, the Knight of the Round Table. One of Albus Dumbledore's middle names is Percival.
The white deer and the Lady of the Lake
To the Celts, a white deer was a messenger from the otherworld. King Arthur followed a white stag but could never catch it, and it became a symbol of a never-ending quest.
In third year, Harry had a dream he was following something white through a forest (PA13). This may have actually been a memory of Voldemort's, since he followed a unicorn through the Forbidden Forest in order to steal it's magical blood (PS15). More likely it foreshadows Professor Lupin teaching Harry the Patronus Charm to protect himself from Dementors. Harry's Patronus was a silvery stag (PA21).
When Harry needed to destroy Salazar Slytherin's locket, Severus Snape sent his silver doe patronus (a deer "made of light") to lead him to a pool in the Forest of Dean where he could see the Sword of Gryffindor like a silver cross at the bottom of the pool (DH19). He tried to dive for it, but nearly drowned due to the Horcrux locket around his neck, and was saved by the arrival of Ron Weasley, who eventually retrieved the Sword himself - continuing the Weasley/King Arthur connection.
Chivalry and the Sword
Chivalry was a code of conduct for knights in Medieval times. On the night Harry was sorted into Gryffindor, the Sorting Hat sang: "Their daring, nerve, and chivalry set Gryffindors apart." That line went through Harry's mind as he thought about diving into the icy pool to retrieve the Sword of Gryffindor (DH19).
Arthur's sword Excalibur was pulled from a stone to show his worthiness, and the Sword of Gryffindor seems to have the same kind of test of character. In Deathly Hallows, Dumbledore's portrait told Snape the Sword of Gryffindor must only be taken "under conditions of need and valor" (DH33), so Snape shouldn't make it too easy for Harry to get it. And indeed, putting the sword underwater in the winter time was certainly a test of "daring, nerve and chivalry."
Inconsistency: the author wrote on Pottermore that Snape put the Sword in the freezing pool "out of spite," as if Snape was making things too hard for the hero (Pottermore: Sword of Gryffindor). First of all, that would assume that Snape wanted to make it harder for Harry to succeed at defeating Voldemort, which really can't be true or Snape wouldn't be out in the Forest of Dean searching for Harry and risking everything by showing his silver doe Patronus to possible Snatchers and anyone else lurking around. For proof of that, Ron saw the silver doe and assumed it was Harry's Patronus stag, and Harry thought Ron had sent the doe (DH19). Anyone could have seen it, and if Snape had been discovered there also it would have been curtains for him earlier than it was. Later in the story the Death Eaters saw Harry's stag and had to be tricked into thinking it was Aberforth's goat (DH28), so the canon is clear that Snape was taking a huge risk by even going there. And to be fair, he seemed to be following Dumbledore's instructions to the letter in making the task challenging for Harry, especially since every other time the sword was used in the canon (by both Harry and Neville) they were in dire straits - Harry was fighting off a Basilisk (CS17), and in Neville's case, the Dark Lord had set his head on fire before Neville got the sword and cut off the head of Nagini the giant snake (DH36). Compared to those circumstance, a dive into a pool of water is like taking a bath.