In the story of Sir Gawain and the Green Knight, the Green Knight visits King Arthur's court and challenges the knights to a beheading game. Sir Gawain accepts the challenge and cuts off the Green Knight's head, but the Green Knight picks up his own head and tells Gawain that he must meet him in a year and a day to receive a blow in return.
After the Green Knight laughs at Gawain's response to his challenge, Gawain offers to do penance for his cowardice and lack of chivalry. He says that he will wear a green girdle around his waist as a symbol of his shame, and that he will do whatever the Green Knight requires of him. This shows Gawain's sincere regret for his actions and his willingness to make amends for his mistakes.
Gawain's offer to do penance highlights the importance of honor, bravery, and integrity in the code of chivalry. By accepting responsibility for his shortcomings and seeking to make amends, Gawain demonstrates his commitment to these ideals and to the ideals of knighthood.
Overall, Gawain's offer to do penance after the Green Knight laughs at his response is a significant moment in the story that showcases Gawain's character and the values of chivalry. Through his actions, Gawain demonstrates that he is willing to admit his mistakes, take responsibility for his actions, and make amends, which are all important qualities of a true knight.