Beowulf and Grendel fight without weapons, and Beowulf, of course, defeats Grendel by ripping his arm out of its socket through sheer strength.
In this warrior society, a leader's skill and strength in war are all important--in part because a leader like Beowulf, still young and untested as a king, must show his retainers that he cannot be defeated by arms (that is, the sword or axe) or by a failure of his strength. Beowulf's decision to avoid using weapons, however, is based on his belief that Grendel is immune to the weapons the Geats normally carry. As he says to Hrothgar:
I have also heard that this wretch/in his rashness recks not of weapons. . . . I therefore disdain to bear a sword or broad shield. . . . (Walton, trans.)
In other words, Grendel is so fierce that he doesn't even need weapons to defeat men, so the use of weapons is, in Beowulf's view, futile.
More important, though, is that during the fight, as Beowulf's men draw their weapons in an attempt to protect him against Grendel, we learn
. . . that no war-blades . . . could even scathe that sinful wrecker;/for he had cast spells against all edged weapons whatever. (Walton, trans., ll. 801-805)
Beowulf's desire, then, to fight Grendel with only his strength--initially because he thinks Grendel is not skilled with weapons--becomes Beowulf's salvation in the battle. Since no weapon forged by man could harm Grendel, man's strength, Beowulf's physical power, is the only weapon by which Grendel can be defeated.