Wow, this is an excellent question.  I have often wondered around Olympics time!  Basically, a sport is active and competitive.  In other words, people are competing against one another in some way that is active and they are moving a lot.

This is a question I have debated with friends many times, never coming to a firm conclusion. As pointed out above, we can broadly define sport as athletic and competitive activity. Yet there are some examples that seem to disagree with this definition.

Many people would probably consider jogging an example of a sport though it is not competitive at all. A contrasting example is walking. Walking is not a sport until we slap the title "competitive" in front of it. Then it's suddenly a sport. 

The first example here of team dancing seems easy to defend as a sport, moreso than the status of golf as a sport mentioned by mwestwood. 

Nowadays, there is bound  to be confusion between games and sports.  Chess certainly is no sport, nor why spelling bees are on ESPN is beyond my ken, too.  Golf, for instance, is considered a sport, but many feel it is  merely  a skill game.

I believe any organized activity with set rules that requires both physical exertion and some sort of competitive desire will qualify as a sport. You will certainly get a good look at the many different types of sporting events during the current Olympic Games, and these are just the summer sports. I'll never forget the debate that occurred when Bobby Fischer faced off against Boris Spasky in their famed chess tournament in the early '70s. Was a chess a sport or not? Many sports news broadcasts included the updates alongside baseball and basketball results, while other stations and newspapers chose to classify it as a non-sporting event.

I would say it involves competition between at least two people or teams, and requires some sort of physical athletic skill. Chess is a competition, but it doesn't involve athletic skill, so it's a game, not a sport.

If it's competitive, and requires physical activity, it could be considered a sport. But really, it shouldn't matter. If you enjoy it, and it helps keep you in shape and challenges you, whether it's defined as a sport really shouldn't concern you. Just keep doing what you're passionate about. The distinctions between what is and is not a sport are so fine as to be meaningless in any case.

Sports are typically athletic activities that require skill, strength, stamina, and physicality.  Most sports involve an element of competition against another team; sometimes the teams interact with each other; like in football, baseball, or basketball, the teams try to score against each other.  In other sports, the competition comes from individuals or teams who compete to deliver the most optimal performance of the challenge, like who can run the fastest sprint, or swim the quickest relay, or perform the most technically accurate dive. 

Competitive dancing has many elements of the aforementioned sports.  It requires skill, strength, stamina, and physicality.  Dancers' bodies have to be in extremely good shape to perform their choreography and often require years of training to reach competition level.  Competitive dancing also involves competition between teams, in which they are scored on the basis of their ability to deliver the most technically challenging, synchronized, and accurate routine.  In that aspect, competitive dancing is really not that different from diving or gymnastics.  Based on these criteria, competitive dancing could definitely be considered a sport.