The Fight Club is an activity where you get to test your ability with your super powers. It is a round-based combat zone where you use nothing but your super powers against a variety of opponents. There's only three instances of it, with only three to four rounds in each one and the final round consisting of a super powered version of a former Saints Row character.
There is only one slight variation to the Fight Club, which occurs on the medium and hard instances of the activity. This is a round called King of the Hill, where you need to stand inside of a green circle until the meter in the upper left completely fills, ending that round.