"The Game" is a rather vague term often used in improv, but it is rarely precisely defined, but often looked for. Here are a few more or less classic definitions. There are others (and if you have a favorite, by all means mail us!)
The game is what is interesting, unique, strange about the scene, the relationship between the characters or the universe in which scene and characters move.
The game is a pattern of interactions.
The "game of the scene" is the interplay of two characters' wants or needs.
Finding the game is establishing a pattern of interesting interactions, and heightening that.
Some will argue that the "game" is the first thing that makes a scene funny. Others will argue that the fun is a byproduct of the game, and as a byproduct, it often strikes us as funny.
Some hold that relationships between characters and "the game" are mutually exclusive. In that resepct, Finding the relationship is about doing slower more involved scene work and finding the Game is quicker and to the point. Others will hold that the 2 are the same, in the following way: one cannot have a solid game that can be repeated in any improv form or format without Relationship. At the same time one cannot make a relationship scene stick without a game to build the scene around it.
In summary, no consensus about the Game and How To Find It.
Be more boring --- Keith Johnstone