New Yevon and their leader, Baralai, think that since the teachings of Yevon before were incorrect that they should try and build Spira into a new. Therefor that makes them "New Yevon."
In serval parts of the game, you will have certain choices concerning New Yevon, 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.