Different games aim to tell different types of stories, and thus are played with different rules.
There are some common elements, though.
Here are some elements you’ll see over and over again:
Playing As Characters
Almost all story games involve the participants creating characters, and roleplaying (pretending to be those characters). For example, in The Shadow of Yesterday you create an adventurer in a fantasy realm. Different games expect a different level of characterization and involvement in the character. Some games expect emotional investment in your character, and for you to act as your character in a pretty serious way. Other games expect you to treat your character as a “playing piece” in the story.
Resolving Conflict Situations Through Rules
Most games are structured so that you build up to a conflict point (perhaps between your characters, or against some external force) and then use a set of rules to resolve the conflict. For example, in Don't Rest Your Head[?] (a game about insomniacs trapped in a phantasmagoric city), when your character encounters dangerous people and insidious forces, you roll a set of dice to determine whether they make it out of the situation okay, and if they face more madness or suffering. Many games rely on dice or cards to randomize the result, which takes the story in unpredictable directions. Often, characters have traits which you’ll involve in the conflict to help you win a desired outcome.
One Person Runs the Game
Often, there’ll be a single player who plays a special managerial role. It’ll be their job to set up conflicts, or perhaps play antagonists, or describe the world existing around the character. Depending on what the game requires, this person will take on different roles to help the people playing central characters to focus on that role. This person is often called the Game Master (abbreviated as GM), or perhaps the Story Guide or something more genre specific. Not every game has a GM though, and we’ve got a list of cool games that don’t in the List of Games.
Using Rules To Structure The Narrative
In general, story games use rules and procedures to structure a particular story experience. For example, in Contenders[?] (a game about boxing, in the style of Rocky or Raging Bull), the rules help structure a story of down-and-out boxers, the trials and tribulations they go through, and the big fights they participate in. The rules and procedures of Ribbon Drive[?], a game about road trips, are all about interacting with the other people on the trip, and learning to let go and enjoy what comes your way.