In the titleThe Most Dangerous Game, "game" is used as the term for any animal hunted for sport. Humans hunt many animals for sport: Ducks, Geese, Bears, Tigers (illegally), Deer, etc. These animals are collectively called game animals, or game. Some game can be very dangerous to hunt e.g. bears, tigers, and leopards, and not just because it may be poaching. All these animals have been known to maul humans, and even kill humans at times.
However, in the short story, General Zaroff hunts a "game" that can fight back in ways he wouldn't necessarily expect: he hunts other men. Rainsford is the "game" and proves to be very dangerous game, killing Ivan, Zaroff's best hound, and ultimately Zaroff himself. That is where the title The Most Dangerous Game comes from.
I agree with tjbrewer's answer, above, that Rainsford is the specific species of "game" that is the most dangerous. But I would add that the title also suggests a second meaning: that Zaroff is actually playing the "most dangerous game" by hunting humans.