Hrothgar asks Beowulf to battle Grendel's mother because she is seeking revenge for Grendel's death, posing a new threat to the kingdom. Beowulf's bravery and skill in defeating Grendel make him the best candidate for facing this new challenge.
Underwater, in a lake.
Grendel's mother kills Aeschere, one of Hrothgar's most trusted advisors, in revenge for her son Grendel's death at the hands of Beowulf.
aeschere
Grendel's mother mourns the death of her son by seeking revenge on Beowulf, who killed him. She attacks Heorot in search of retaliation, ultimately resulting in a battle with Beowulf himself.
Grendel's mother steals the head of one of Hrothgar's most trusted advisors, Aeshere, from the great hall.
Hrunting, a sword handed down from Hrothgar's forefathers.
A swamp
After Beowulf kills Grendel's mother, he sees Grendel's lifeless body and beheads it as proof of his victory. He then finds the decapitated head of Grendel and takes it with him as he returns to Heorot to show it to Hrothgar. The Geats and Danes rejoice in Beowulf's triumph over the monsters.
While Grendel's mother is hard for Beowulf to defeat, he IS able to use a sword, where as he has to battle Grendel bare handed.
he dies later on
Beowulf kills Grendel's mother by using her own weapon, a giant's sword, against her. This is ironic because he defeats her with the very weapon she was planning to use against him. Additionally, he avenges Grendel's death by killing his mother in the same manner that Grendel killed Hrothgar's men.
Beowulf accepts to go after Grendel's mother but first sets a number of conditions. If Beowulf is to be killed, Hrothgar would have to take care of Beowulf's family and Unferth would inherit his possessions and estate.