On the most basic level, the figure of the Green Knight represents an insurmountable challenge; he entreats King Arthur's court to fight him on the promise that any knight may chop off his head, but in the following year, they must fight again and he will behead the knight. The entire court is so cowed by his proposal that Arthur is on the verge of accepting when Gawain volunteers.
On a deeper level, the Green Knight represents regeneration; he has an uncanny ability to restore himself, even to come back from what should be mortal injuries. Scholars equate the Green Knight with the "spirit of vegetation. Trees can live far longer than human beings, and they have regenerative powers that people have always envied" ("The Figure of the Green Knight," eNotes). His natural ability and regenerative powers combine in force to make a foe both daunting and formidable for Sir Gawain to overcome; in this way, Gawain's heroic deeds seem even more impressive.