It's a mathematical theory about the optimality of strategies in a game. A game is generally defined as a set of rules that determine the possible actions that each player can make and the rewards given to each player in a round of play (each player makes a move). John Nash is best known for the "Nash Equilibrium" which is a theorem that defines a set of conditions by which the strategy of each player is locally optimal given the strategies of the other players. More interesting, however, is that the global optimum (most rewards for all) might not be one of those Nash equilibrium points, from which you can prove that cooperation is always better than competition.
This theory has implications in economy, sociology, ethics, etc.. as it provides a mathematical framework to evaluate the optimality and stability (stable equilibriums) of a collective set of strategies. For example, the stock market can be seen as a game in which all players try to make the best investments (actions), and depending on each others investments, they get rewards (returns on investments). And so, being able analyse which moves are the best in that context is quite important (for the investors), and also, analyzing what should be the best collective investment strategies is quite important (for the society at large). In ethics, there are similar considerations, i.e., maximizing the well-being of everyone.
Now, if you think that this has much to do with computer games or things like that, then you are mistaken. Game theory could have reprecussions on computer games (e.g., what is the best strategy when playing the game), but in general, computer games are far too complex for this theory to be applicable beyond some very high-level analysis of it. I think that some researchers have made some analysis of simple games (e.g., checkers), but that's about it. Also, pretty much everything in game theory is intractable, meaning that it is far too complex to ever really be able to calculate anything in any reasonable amount of time, meaning that you wouldn't be able to use this to, say, compute the optimal strategy for a player (bot) in real-time.