1. Play, from the opening breakshot until one player has won (or the game has been halted for some reason by a referee). Games are the units that make up matches, races (in some senses of that term) and rounds. Essentially the same as frame, except with regards to straight pool, which is a multi-rackgame.
2. An identifiable, codifiable set of rules. pool is not a game, but a class of games. Nine-ball is a game.
3. Note: There are also slang usages, such as "to have game" (to be a good player, as in "he['s] got game") and "to be game" (to be willing to play or to gamble, as in "yeah, I'm game, so let's see what you've got"). But these usages are not particular to cue sports.