Interaction has often been used as a term to support the argument that games should not be studied with the concepts and tools of narratology. In this article I go the other way and approach interaction from a narratology-based perspective.
The Layers of a Narrative
An analyses of narrative structure reveals four narrative layers:
Events (or Functions): all actions that are carried out to initiate, continue and terminate a logically connected sequence [an example from Diablo: zombie is on attack --> kill zombie --> collect gold]. 
Story persons (or Actants): the fictional beings who carry out the actions which articulate as events. Often they'll signify something larger than their partiular presence and connect to a "will". [Back to the Diablo example: All monsters in the game are actants or story persons. But ultimately they articulate under a narrative "force" that we can identify as "Diablo". This is the "will" that aims to break the "will" of the story person that we act as in the game world.]
Storytelling (or Narration): the various techniques and methods through which the events and story persons to which they belong to are presented to the player. [Examples from Diablo: Exposure of information through entering dialogues with the people in town; gothic iconography and low gamma to foster mood and to communicate genre, isometric view, player's in-game presence visually divided into avatar and cursor etc]
Narrative Situation: the broader rules and conventions that shape the way in which games (game narratives) are consumed. This is not really part of the narrative itself, but rather means the circumstances and cultural codes that allow the game narrative to be perceived and consumed as such.
The Archetype of Interaction in Video Games: Influencing the Events Layer
Historically, the video game has been a form of narrative that through input allowed players to influence how a row of Events will turn out. While players were not the creators (authors) of the game universe, they were nevertheless the Story Persons that the design of the control and interface (and the choices they made through these) amounted to . This is the archetype of video game interactivity and it remains until today fundamental to any game. Gameplay built around influencing the Events layer has been (probably because of technical constraints) the dominant mode of interaction for many decades, exemplified right from the beginning through games such as Spacewar!, Pong, Asteroids, Space Invaders, Centipede, Pac-Man, Donkey Kong, Zaxxon etc etc etc.
Traditionally, the author/game designer would narrow down the possibility space of the game in order to maintain a certain degree of logical continuity within the Events. Despite her freedom, the player would stay on 'track' because the design of the 'system' and its mechanics would exercise its structuring force onto her choices and behaviors. Today however, we witness games in which the possibility space in regard to the Events layer is kept broader, especially in games that want us to consider the ethical implications of our choices and present to us more than one major course of action. Games like Fable, Deus Ex and Black & White, known for their 'non-linearity' and multiple endings, come in mind.
Expanding Into Other Narrative Layers: Story Persons and Narration
Over time we saw game designers explore the possibilities of the other layers of narrative. MUDs and genres such as RPGs and Simulations put emphasis on a variety of interactions that had as their subjects story persons and other game world existents: Players could modify or remove existing characters, create characters from scratch, change their looks, traits and behaviors. Other games allowed us to change the environment, to add, modify or remove objects, to modify terrain etc. Civilization, Sim City, Diablo, The Sims, Rollercoaster Tycoon, Railroad Tycoon, X-Com Apocalypse, Football Manager are games that come in mind. They added onto the archetypical layer of events, the ability to modify the layer of story persons and existents.
Nowadays, most modern games have added to these two layers the modification of the layer of Narration: We can now modify the camera angle or behavior, switch between POVs, chose the music and gamma (ambience) that we want to set the tone of the game world while we play, alter the frequency of in-game commentary, switch on or off replays etc etc etc... Pro Evolution Soccer, Need for Speed, Medal of Honor are just a few examples that come in mind.
New Narrative Situations and the Future of Interaction
With Project Natal announced, we prepare ourselves to witness a new type of narrative situation (our broader defined relationship with the medium that we interact with). The way in which we will 'consume' games seems like it will undergo significant changes. However, we yet have to see how "the player as controller" will gain functionality and what constraints it will bring with it in terms of interaction with the other narrative layers present in a game.
 The Events layer implies that characters in a story are in a constant process of decision-making. To give an example: [phone is ringing --> Option A) answer call; Option B) don't answer call --> different outcomes for Option A and B]. In extreme situations chosing the 'wrong' option could mean "end of story".
 In drama theory it is often pointed out that choice is character.