There are so many swords in Le Morte D'Arthur, you'd be forgiven for losing track. The first important sword that appears is the well-known "sword in the stone" that Arthur pulls in a churchyard, which proves that he is king of all England.
From this example, you might think that a sword represents power… and you're not wrong. After all, Arthur and his knights gain and maintain power "by the sword." Arthur wins wars against an alliance of Northern kings and against Lucius, Emperor of Rome, by the sword, on the way to solidifying his kingship. Plus his knights defeat other knights with swords, and send those guys to Arthur as vassals. So if a sword represents power, it also represents the way that Arthur and his knights hold onto that power, through violence.
But the sword that Arthur pulls from the stone also tells us something about his identity, too. When Arthur pulls the sword from the stone, it tells him and everyone else that he's the real deal: the King of England. Plus, other swords give us other identities, too. In the "Tale of Balyn and Balan," for example, a damsel arrives in Arthur's court with a sword that can only be pulled by "a passynge good man of hys hondys and of hys dedis, and withoute velony other trechory, and withoute treson" (40.41-42). Balyn pulls it, and – bam – he's a man with these attributes.
Galahad also pulls a sword in a stone that confers an identity on him, this time as the best knight in the world. In the same story, Galahad becomes the owner of a sword with a history that stretches back to the Old Testament, establishing his identity as the heir to an ancient and mysterious holy family. So when a man in Camelot claims and wields a sword, it tells us a lot about what makes that man worthy.
As it turns out, armor in general is pretty important to one's identity, and the significance of the sword is a part of that, too. Remember that a knight is recognizable on the battlefield only by his armor, which leads to a whole lot of unfortunate cases of mistaken identity for knights who lose or swap theirs. A knight's coat of arms, or the markings on his shield, tells everyone who sees it what family he belongs to. His sword, on the other hand, tells others who he is as an individual, and his role in Camelot.