First off, the character Beowulf is known for his exceeding bravery. This is evident because upon first hearing that Grendel is attacking Hrothgar's mead hall, Herot, Beowulf immediately sets out to go and defeat Grendel even though Grendel has killed everyone else who has come to try to kill him. Additionally, Beowulf determines that when he fights Grendel he won't use a sword but rather will get the job done with his bare hands. He asks himself the retorical question which loosely paraprased goes something like this: should I send my sword somewhere where I'm afraid to go?
Beowulf's bravery, however, can also be viewed in another life: that is that he's too overconfident.
Grendel has more striking character trait that shines though and that is his bitterness towards humankind and their special relationship with God. Apparently Grendel is upset that God has cursed his ancestors and now he must live a miserable existance. (Go figure!)
Grendel's mother has a similar type of feature to her son: she's out to get revenge. Beowulf kills her son, and like any mother, she's out to get revenge for her son's death. She wants a life for a life.