Overview
A Nash Equilibrium exists when all players in a game have chosen their strategies, and any individual in the game, given a choice to be able to change her strategy after seeing the strategies of her competitors, she decides to keep her strategy. (That is, there is no regret)
Nash Theorem
Nash's theorem asserts the existence of at least one Nash equilibrium if there are finitely many players who can choose from a finite set of pure strategies, when mixed strategies are allowed.