Popularly, several games have been reclassified as "mind sports"--things like video games and even poker are now called sports. By this standard chess is a sport.
But everybody is wrong. 
In fact, I'd go so far as to say a lot of games people have traditionally thought of as sports should be reclassified as "physical games of skill" or "physical competitions" and reserve the term sport for a more specific set.
Before "mind sports" became a thing, chess was definitely not a sport. It was a game of skill. With a couple possible exceptions: Chess boxing (which is new) and bullet/blitz chess. Chess boxing is obvious because it combines chess with a bona fide sport. Bullet chess, though, required physical skill--rapidly moving pieces and hitting the clock faster than your opponent.
Yet, clearly even bullet chess is different from sports like basketball, football (all kinds), or even baseball. And that difference is where you can clearly define what a sport is in contrast to everything else. In basketball, football, and baseball your actions are opposed directly by the physical actions of other players. In chess, your physical action is not opposed directly by the opponent. They aren't trying to physically restrain you from placing your piece or put their piece on the square faster than you can--they have to wait for you to make your move.
Many games of skill are like this: Darts, golf, and billiards come to mind. These games all require extremely high levels of physical skill--but they are not the same as sports. There is no opponent opposing your actions as you make them.
Competitions like track and field, diving, and figure skating also lack this feature and are also not games. These require the same (and sometimes even greater) athleticism as sports, but belong in a different, but no less respected category.
Games of the mind like chess, poker, and the like deserve respect for the skill needed and get credit for the amount of endurance required for high level play--but they are not the same. They are their own category and should be respected for what they are.
Of course, if we want to go back to the Old French "desport" from which our word derives then any sort of frivolous leisurely play would be sport and even posting on forums could be considered a sport.
Chess falls into the category of "mental sports" or "mind sports" which are competitive in nature because they require strategy, skill and sharp thinking.