Here is an exceedingly narrow definition sure to provoke some arguments:
A sport is a contest of physical prowess characterized by scoring with symmetrical objectives, offense and defense, with the purpose of entertainment. This excludes all marksmanship events, races, and indirect competitions. Track and field, golf, basically anything you can have a world record for doesn't count because if defense was meaningfully part of the competition, we wouldn't be able to compare achievements.
Some typical sports:
football, american football, tennis, rugby, basketball, baseball
Some surprising not-sports:
golf, all track and field events, nascar, darts, snowboarding, motocross, ballet, cheerleading, competitive dance, american idol, swimming, synchronized swimming
Some surprising sports:
croquet, ping pong, pool & billiards, water polo, quidditch (albeit a sport with terrible and ridiculous rules), fencing, starcraft
Please note that this is not a value judgement. I don't think croquet is more awesome than motocross or the 100 meter dash, or that golf players are less skilled than fencers, or that swimmers are less athletic than quarterbacks, I just think that my definition of sport usefully describes a similar set of physical contests.
Ready, set, irrelevantly disagree with me!