First of all, I’m going to make fun of the cover of the Legion of Super-Heroes Archive Volume 1. It looks like the heroes on the edge have dorky little capes that are blowing straight up in the air. (Apparently, that shape behind them is the Legion Clubhouse.)
The Legion first appeared in 1958 in a story about Superboy, who is Superman as a kid. The Legion members reveal to Superboy that they are superheroes from 1000 years in the future. So, they’re from the year 2958. Since all of the Superman franchise comics are supposed to take place in the current year, the Legion is now from the year 3010. Next year they’ll be from 3011, and so on and so forth.
The Legion of Super-Heroes has many members with strange powers, including Lightning Lad and Saturn Girl. Every year, the Legion accepts one new member, and they want to induct Superboy. They bring Superboy to the future, and they act like the Legion of Super-Douches to him. They make him take many tests to see if he should be in the group, but every time the challenge begins he has to run off and save someone, so he never gets to show off his skills. The Super-Douches then tell little Superboy that he’s not good enough to be a Legion member, and they make him cry! Then, they tell him it was all part of the hazing and let him join.
After Superboy is part of the Legion, the stories unfold. One of the most popular themes includes Legion-traitors. There are also a lot of robots and time-traveling because those stories sold well. The Legion also has life sized statues of themselves in their clubhouse because they're egomaniacs and it helps keep their names straight since there are so many of them!