In Game of Thrones, the bedding tradition is known as the bedding ceremony. It is a custom in which a newly wed couple consummates their marriage by spending their first night together in bed. The couple is usually required to strip down to their birthday suits and spend the night naked. This ritual is meant to symbolize the couple's new life together and their willingness to be completely vulnerable with each other.
The bedding ceremony was first introduced in the show's pilot episode, when King Robert Baratheon and his new queen, Cersei Lannister, were shown consummating their marriage. The scene caused quite a stir, both because of its graphic nature and because it revealed that Cersei had been cheating on her husband with her twin brother Jaime.
Since then, the bedding ceremony has been mentioned a few times on Game of Thrones, but it hasn't been shown onscreen since that initial scene. In one case, Sansa Stark was forced into a loveless marriage with Tyrion Lannister and was dreading having to go through with the bedding ceremony. However, Tyrion ended up being surprisingly kind and gentle with Sansa on their wedding night, putting her mind at ease (at least for that evening).
More recently, Daenerys Targaryen refused to participate in the bedding ceremony when she married Khal Drogo in season one. However, she eventually changed her mind after realizing that it was important to Drogo and his people.
While the bedding ceremony may seem like a strange custom to modern audiences, it's actually based on real-life medieval customs. In those days, weddings were often more about political alliances than love, so families would want to make sure that couples went through with consummating the marriage. By making them do it in front of everyone (and without any clothes), there would be no doubt that they had indeed slept together.
So there you have it: the history and meaning behind Game of Thrones' infamous bedding tradition!