Symbolism, Imagery, Allegory
There are several different famous swords in Beowulf – so many, in fact, that you might have trouble keeping them straight. First, there's the sword that Hrothgar gives Beowulf after he kills Grendel (1022). Second, there's Hrunting, the sword that Unferth lends to Beowulf to fight Grendel's mother (1458). Unfortunately, Hrunting fails to do any damage to the monster, so Beowulf grabs another sword from her horde of treasure (1557). This third sword decapitates her, but the blade melts when it touches her poisonous blood. After he brings the hilt back to the surface, Hrothgar discovers it is covered in engravings of the great flood and destruction of the giants. Fourth, there's a gem-studded sword that King Hygelac gives Beowulf to celebrate his great deeds (2193). We can probably assume that this is the sword called Naegling, which breaks when Beowulf tries to use it to kill the dragon (2680).
There are a few other swords in the story here and there, but these four are the most important ones: Hrothgar's gift, Hrunting, the sword with the engraved hilt, and Naegling.
That leaves us with two questions: why are there so many different swords in the epic, and why do they so often fail to harm the enemy? And no, we don't think the answer is just that swords are phallic symbols. Well, there are many different swords in the epic for a pretty obvious reason – in a warrior culture, weapons are going to be pretty important, and there are a lot of them around. There are also different weapons for different purposes, and sometimes one sword can succeed where another fails, depending its quality and its history.
But the really strange thing in Beowulf is that, frequently, swords don't do their job. Hrunting won't cut Grendel's mother; Naegling snaps when Beowulf swings it at the dragon; the sword with the engraved hilt melts in Grendel's mother's blood. We're getting an inkling that the poet wants to remind us of the futility of battle. It also seems that Beowulf does better when he uses his own body strength against the monsters around him, instead of weapons, which are almost like cheating because they give him an artificial advantage. Alternatively, at one point, the narrator suggests that Beowulf is so strong that his mighty strokes break blades in half, so perhaps Beowulf's heroism is greater than mere weapons could make it.