The green girdle was given to Gawain by Lady Bertilak. She gave it to him as a lover's token. She also mentioned that the wearer will become invincible.
For Gawain, it is a symbol of his survival at the expense of his code of honor. When he failed the Green knight test, he wore his green girdle to symbolize his failure and be a reminder for him of a lesson learned.
When the Knights of the Round Table wore green girdle as a sign of solidarity, it symbolizes the men's honor.