The stories of King Arthur and the Knights of the Round Table have been told for many centuries. These tales represent the gold age of chivalry and honor. The knights in these stories never intentionally harm the innocent and almost always obeyed their code of honor. In the story of Gawain and the Green Knight, however, even Gawain violates a few of the rules. I think this story represents the slow demise of chivalry.
The first example of the lack of chivalry in the Round Table happens when the Green Knight rides into the room on his horse and challenges any man to trade blows with him. Such an insult would have been met with many challenges at the peek of chivalry, but only King Arthur and Sir Gawain took him up on his challenge. One also gets the impression that Gawain did this only to prevent his king from taking unnecessary risks with his life. Although he manages to decapitate the Green Knight, he doesn't kill him. A sense of dread becomes apparent as he realizes he is bound by his word to have the same fate as the Green Knight's body in one year and a day.
Another example of the demise of chivalry occurs at the Green Knights castle. Sir Gawain manages to keep his word for two of the days, but on the third day, he keeps the lady's scarf. The reason he does this is obviously for its protective properties. This seems like a good idea, but this violates his promise to give everything he gets back to the lord of the manor. It also violates his faith in God's ability to save him from being decapitated. Sir Gawain isn't as good at following the code of honor in this story as in the other stories.
The final example I will point out is when he faces the Green Knight. He is confident that the Green Knight can't kill him, just like he couldn't kill the Green Knight a year ago. However, he gets impatient with the Green Knight's childish tricks with the axe, making feints at his neck and insulting him when he flinches. After the third cut, which cuts him on the back of the neck, he leaps to his feet and challenges the Green Knight, insulting him the whole time. Chivalry would have suggested that he should have been more polite to the knight. Gawain chooses the easy path again and lets his aggression take control of the situation. This was disappointing to me.
But however disappointed I may have been at that point, I had to admit that he wasn't as bad a knight as I had assumed. When he learned that the Green Knight was testing him, he repented his past acts and wore the green scarf for the rest of his days as a result of this. The story represents the slow lingering demise of chivalry rather than the death of it.