You can play any game and make up a story to go along with it. This can be a lot of fun - try actually narrating the events in the two warring countries represented in chess sometime - but it isn't a story game.
A story game is a game in which the players create fiction, and the fiction can in turn influence the play of the game.
This crucial difference (which we drew from the writings of RPG theorist Brand Robins) can express itself in a game's design in a number of ways. The most common way so far happens to be the way that was invented when story gaming was invented: the role-playing game. This gets tricky, because the term "role-playing game" has never had a firm definition across all of its communities of players, and because many different people have attachments to mutually incompatible meanings for the term. But we'll have a go anyway: in most role-playing games, one player takes the position of gamemaster, and the others each control a fictional character and its actions. The loop of "player says what his/her character does, gamemaster describes how the world and non-player characters respond, player takes a new action in the new situation" is a very direct, visceral and rewarding way for game-created story to influence gameplay - so direct and rewarding that it's taken 30-some years for very many others to come to light.
This site, however, is mostly about those other ways. The possibilities of the fusion of story, game, and collaboration are vast and underexplored - just because we have one great way to bring them together is no excuse not to look for others.