"Game" is any situation in which there are several decision-makers, and each of them wants to optimize their results. The optimizing decision will depend on the decisions of the others.
Game theory is a branch of math built around something called games. There are multiple players, and they each have some strategy to try to get the most points. You try to find something called the Nash equilibrium, where each player's strategy is the best possible strategy given their opponent's strategy.
Focuses on how people cooperate/compete and make decisions. Its a paradigm that usually puts participants in the situation where they have to play a game of resources and cooperation/competition.