paulflexes saidA sport is any activity that can be used as a competition.
Playing the piano? Bodybuilding? It's a good question but there's no real answer. The only time I worry about it is when it comes to the Olympics. What's the deal with the girls dancing around with batons and streamers?
I'm kind of bothered by any sport that uses judges; e.g., gymnastics, diving, synchronized swimming, cheer leading, your baton and streamers example, figure skating, etc. On the other hand, even with sports that don't use judges you have referees, umpires, etc. which are somewhat similar to judges.
But ultimately, competitive sports are fundamentally there to provide entertainment for the spectators. Given that they're merely entertainment, I don't think it's important to worry about whether or not something is a sport.