Background of "Sir Gawain and The Green Knight".
I am assuming that you are asking for the "pre-story" behind this piece of literature. According to this story, Gawain is one of the knights among Arthur's crew. They are in the midst of a meeting when The Green Knight appears and challenges Arthur. Gawain stands up to take the challenge in place of Arthur which shows his loyalty, honor, and bravery.
Now, the reader comes to understand later that The Green Knight has come to teach Arthur and his knights a lesson. Apparently, before this story takes place, events have occured which are evidence that the knights of the Round Table have been slacking in their chivalry--honor, bravery, generosity, protection of the weak, etc. We don't have a whole lot more than that as far as background for this particular Arthur legend.
At any rate, Gawain is sent on a quest which lasts for one year where he is to seek and find the Green Knight's home and allow him to cut off Gawain's head as Gawain did when the Green Knight visited Camelot. In the process, Gawain is put to a test while staying at the home of a country gentleman (who, unbeknownst to Gawain, is The Green Knight) which includes trading whatever the other has received during the course of the day each evening. The gentleman goes hunting, Gawain is left at home with the gentleman's wife who attempts to seduce him and gives him kisses as well as a green sash which will protect its wearer from harm. Gawain gives the gentleman the kisses, but keeps the sash as he is afraid of the Green Knight's axe--Gawain will not be able to pick up his head, tuck it under his arm, and ride off as the Green Knight did on their first meeting.
The Green Knight chastises Gawain for not telling the truth about the sash and for flinching when he first swung his axe. In the end, the two knights are friends and leave each other's company with plans to meet each other for New Year's meals in the future. Gawain returns to Camelot to educate the rest of the Round Table knights about his lesson and to reiterate their pledges to exercis chivalry at all times.