The Youth League is led by Nooj, an old crusader friends with Paine, Baralai, and Gippal. Members of the Youth League are doing their best to keep Spira the way it is. Nooj and the members of Youth League do not agree with all of the things New Yevon does, and they never seem to get along.
In serval parts of the game, you will have certain choices concerning the Youth League, and you will have to decide which group you will ally with, New Yevon or the Youth League.
Depending which side you chose will determine how the story of the game goes. It also determines how much completion percentage you will aquire for that section of the chapter.