Gawain's quest is to fulfill the terms of the challenge he accepted from the Green Knight. At the start of the poem, the Green Knight visit's Arthur's court and challenges Arthur to deal him a blow, with the understanding that in a year, Arthur will travel to the Green Knight's castle and accept a similar blow.
Gawain is good Knight in the sense that he—mostly—does what is expected of knights. I think a larger question the poem poses for modern readers is whether these expectations are reasonable. Gawain is admirable in his acceptance of the Green Knight's challenge on behalf of Arthur, and in his honoring his pledge to seek out the Green Knight at the Green Chapel a year after he deals him what ought to have been a killing blow. His virtue is without question, but also suggests that he is a bit naive.
His encounter with with Lord Bertilak and his lady tests his virtue. Lady Bertilak's attempts to seduce Gawain are not successful, and in fact Gawain lives up to a promise to share...
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