I would say I have to agree and disagree with can_duathlete. As a professional dancer I do not think that dance is a sport. It's an art. There's also recreational dancing that we all do, but still I don't think of that as a sport.
There are dance competitions held all over the world, and there's different competitions for every style of dance. You have jazz, ballroom, latin, ballet, contemporary, and other competitions. However, the key thing here is that in each one there is a score for artistry. I feel like that can be very subjective. Everyone likes something different about someone's dancing.
On the other hand I do think dancers are athletes. Just like athletes, dancers are required to perform precision moves at a very high level of performance, and it's expected to be the same every time. And yes, some people are faster movers and higher jumpers than others. Every dancer has their forte in their field of dance. The trick is making everything look as good as the things that come easily to you.
As for the rules of dance...yes there is no 4th down or innings like sports. But there are rules within the techniques of different dances. If it's wrong it's wrong. In ballroom you have the placement of the partners arms. In ballet you must maintain proper body alignment. If you watch So You Think You Can Dance you'll hear all of these things all the time.
Anyway, bottom line is; Dance is not sport, but dancers are athletes.