Game theory seldom deals with games that people actually play, such as Chess or Poker. Instead, simplified, abstract situations similar to those that arise during real games are studied: there are two or more players that have different options at each turn. The payoff for each player is determined by the choices that all players make, often independently.
A classical example for the simplest type of games, two-player zero-sum games, is rock-paper-scissors. Here players have the same choices in each round, decisions are make simultaneously and independently, and one player's gain is the other player's loss - hence the name zero-sum game.
A mathematical analysis shows that, against a rational opponent, there is no better way of playing than to choose rock, paper, and scissors at random with a probability of 1/3 each. The rational opponent will do likewise. John von Neumann showed in a groundbreaking analysis that such an equilibrium of strategies exists for each two-person ZSG, in which no player can improve his expected payoff by unilaterally deviating from his stragey while the other player sticks to his. Often a so-called mixed strategy, in which some of the available options are chosen at random with given probabilities, turns out to be optimal.
This is one of the crucial points of game theory: each player wants to maximize a function that depends on quantities outside of his direct influence, which can, however, change in response to his actions - namely, the strategy of his opponents.
Things get more complicated if the assumption of a zero-sum game is dropped, and each player has his own payoff matrix. One infamous example is the "Prisoner's Dilemma": both players have the options of "cooperation" or "defection". If both cooperate, both share a given payoff. If both defect, they get a smaller, but not catastrophic payoff. However, if one offers cooperation while the other defects, the defector gets the maximum payoff, while the cooperator pays dearly. The sobering analysis shows that if the game is played only once, there is only one rational action - defect, betray, cheat. Cooperation can only arise if the game is repeated several times.
Many analogies to the prisoner's dilemma can be found in everyday life: pay for the ticket or steal a ride? If everyone stole a ride, there would be no more public transportation. Governments during the cold war had to think about the prisoner's dilemma as well: should one start a war that brings high losses, or wait and hope that the opponent does launch a surprise attack that would cost even more dearly?
In biology, game theory has revealed interesting aspects of evolution. For example, why do trees have such high stems? Wouldn't it be more economic if all agreed on a smaller height and saved the material necessary for growing so tall? Indeed it would, but if a mutant tree develops a higher stem than its competitors, it gains an advantage that enables it to have more offpring - which carry the "high stem"-gene.
Ways of behavior can therefore only exist for long times (be evolutionarily stable) if deviating behavior gives no advantages in the given environment. This kind of analysis has been applied to may aspects of human behavior (from mating to warfare) and gives many insights into why seemingly irrational behavior can pay off in the long term.