A game, in its simplest terms, is a list of possible "moves" that two players, called left and right, can make. Every move is in fact, another game, such that each game can be considered a single state that the game can exist in.
Combinatorial game theory (CGT) is a mathematical theory that only studies two-player games which have a position which the players take turns changing in defined ways or moves to achieve a defined winning condition. CGT does not study games of chance (like poker), but restricts itself to games whose position is public to both players, and in which the set of available moves is also public. CGT principles can be applied to games like chess, checkers, Go, Hex, and Connect6 but these games are mostly too complicated to allow complete analysis (although the theory has had some recent successes in analyzing Go endgames).
Applying CGT to a position attempts to determine the optimum sequence of moves for both players until the game ends, and by doing so discover the optimum move in any position. In practice, this process is tortuously difficult unless the game is very simple.
CGT should not be confused with another mathematical theory, traditionally called game theory, used in the theory of economic competition and cooperation. Game theory includes games of chance, games of imperfect knowledge and games in which players move simultaneously.
The bible of combinatorial game theory is Winning Ways for your Mathematical Plays, by E. R. Berlekamp, J. H. Conway, and R. K. Guy; the mathematical foundations of the field are provided by Conway's earlier book On Numbers and Games.
and, believe me CGT is never complete with so much stuff around to know about.
some great links:
Combinatorial Game Theory
Introductory combinatorial game theory
