The Need for Nonprofits

I have a question about nonprofits. Is it true that all nonprofits pay nothing in taxes? And if so, why is that allowed? I understand it’s nice to give charities a break on their taxes, but why are we giving big tax breaks to huge organizations that pay their CEOs tons of money? If employees are making big bucks, doesn’t that mean the organization has plenty of money? And why are some organizations that are clearly not charities allowed to be nonprofits? The NFL is a nonprofit, and that seems really ridiculous to me: they make tons of money and should be taxed a lot! Even colleges, it seems to me, make a lot and should be taxed. Experts, can you explain this to me?