The world of Beowulf is one built on oral tradition.  In the poem, this is reflected in the importance the characters assign to being remembered (after they die gloriously in battle, presumably).  The importance of reputation surfaces throughout the poem, but it is when Beowulf arrival at the hall at Heorot and in his conversation with Unferth that the importance of reputation is established.

When Beowulf arrives, he does not present himself as if his being respected is a foregone conclusion.  He introduces himself not by simply stating his name and his place of origin, but by providing a kind of resume of his exploits.  Unferth, hearing what Beowulf says and the reactions of those around him to Beowulf's words, challenges his reputation.  He says:

Beowulf's quest,

sturdy seafarer's, sorely galled him;

ever he envied that other men

should more achieve in middle-earth

of fame under heaven than he himself. -

"Art thou that Beowulf, Breca's rival,

who emulous swam on the open sea,

when for pride the pair of you proved the floods,

and wantonly dared in waters deep

to risk your lives? No living man,

or lief or loath, from your labor dire

could you dissuade, from swimming the main.

Ocean-tides with your arms ye covered,

with strenuous hands the sea-streets measured,

swam o'er the waters. Winter's storm

rolled the rough waves. In realm of sea

a sennight strove ye. In swimming he topped thee,

had more of main!

Unferth actually has the nerve to challenge Beowulf's account of his exploits.  Unferth questions Beowulf's reputation, something that must be addressed immediately and decisively by Beowulf.  To this end, he responds to this challenge, saying:

"What a deal hast uttered, dear my Unferth,

drunken with beer, of Breca now,

told of his triumph! Truth I claim it,

that I had more of might in the sea

than any man else, more ocean-endurance. 

[...]Together we twain on the tides abode

five nights full till the flood divided us,

churning waves and chillest weather,

darkling night, and the northern wind

ruthless rushed on us: rough was the surge.

Now the wrath of the sea-fish rose apace;

yet me 'gainst the monsters my mailed coat,

hard and hand-linked, help afforded, -

battle-sark braided my breast to ward,

garnished with gold. There grasped me firm

and haled me to bottom the hated foe,

with grimmest gripe. 'Twas granted me, though,

to pierce the monster with point of sword,

with blade of battle: huge beast of the sea

was whelmed by the hurly through hand of mine.

Beowulf is determined to set the record straight, because any threat to his reputation would be disastrous.  He cannot fathom being remembered for being a liar or as someone who accomplished nothing honorable in life.  Throughout the poem, Beowulf and the other characters continue to acknowledge the importance of reputation to their own lives but also to those of their families.  Similar examples can be found throughout the poem.

I'll explain the idea of reputation in Beowulf and give you an example or two, and leave the rest for you to find on your own.

The idea of gaining immortality, of a sort, by being remembered is at least as old as the Greeks.  It is vital in Beowulf, as well.  One is remembered for having done great deeds, and heroes want to be remembered.  This is their way of gaining immortality. 

Near the close of the epic poem, Beowulf asks Wiglaf to have a tower built for him:

"...Have

The brave Geats build me a tomb,

When the funeral flames have burned me, and build it

Here, at the water's edge, high

On this spit of land, so sailors can see

This tower, and remember by name, and call it

Beowulf's tower, and boats in the darkness

And mist, crossing the sea, will know it."  (823-830)

Beowulf's dying wish is to be remembered.  That's what motivates him throughout the poem.