Gawain wears the green girdle as a token of his failure for not keeping his promise. He was supposed to give the host that he would give him whatever he gained that day, but he didn't give him the green girdle for fear that he would die. After being called out because of that, Gawain swears to wear the green sash for the rest of his life, and the Knights of the Round Table start wearing it as well after absolving him of blame.
