As described in Chapter XXII of Beowulf, Beowulf first encounters Grendel's mother.
Then the lord of the war-like Geats who did not shrink from combat seized Grendel's mother by the shoulder; that fierce one filled with rage then flung his deadly foe, and she fell to the ground. She swiftly paid him back with her grisly grasp, and grappled with him. Spent with struggle, the warrior stumbled—that fiercest of fighters fell. She hurled herself on the hall's visitor, and drew her broad, brown-edged knife to avenge her only son. The braided mail about his breast prevented death, and barred point and blade from entering.
Although he comes thinking he is prepared (with the heirloom sword, Hrunting (known for never losing a battle)), Beowulf is flung off of her easily. Grendel's mother tries to stab Beowulf, but the mail (braided metal worn for protection) did not allow the blade to cut. The following passage is found at the opening of chapter XXIII.
The among the battle gear there he saw a blade triumphant—an old sword of the giants, an heirloom of warriors, a peerless weapon. It was larger than other men could carry into battle- bandying, as it had been wrought by giants. Then did he, champion of the Scyldings, grasp the hilt of the blade and brandish the sword; heedless of his life, he smote so ferociously that it caught her by the neck, breaking her bones. The sword pierced through the flesh of that doomed one; she fell to the floor. The sword was gory, and the man was pleased with his work.
Beowulf, spotting a large sword (which the narrator believes has been made by giants), grabs the sword and attacks the monster. The blade hits Grendel's mother in the neck, breaking her neck, and cutting off her head.
Beowulf is only able to defeat the monster by using a weapon with parallels her (being supernatural). His original weapon, made by human hands, is no match for the monster.
After killing Grendel's mother, "Then light shone forth; it was bright in there, as when heaven's candle shines in a sky without clouds." This illustration shows that the battle with Grendel's mother was one of epic proportions (good over evil), given the light of God shines after Beowulf's victory.
After finding Grendel's mother in a cave, Beowulf finds it extremely difficult to kill her, especially after failing to pierce her skin with his sword, Unferth's Hrunting. He decides to fight with bare hands, trusting his strength. She tries to kill him with a dagger, but Beowulf's armor is strong enough to resist the attack this time. Beowulf manages to throw her off and grabs an enormous sword he spots. It is extraordinarily decorated, and suggests something of "an old sword made by giants, string of its edges, glory of warriors." Beowulf attacks from above her and succeeds in breaking her collar-bones and slicing her into two. When he sees the body of dead Grendel in her den, he cuts its head off to present to Hrothgar. When he does, the lake becomes red with blood and the people waiting interpret this as Beowulf's death.
The blade of the sword starts melting because the monster's blood is too hot and poisonous. Beowulf swims ashore with only the monster's head and the handle of the sword, and is greeted by his fellow warriors.