Wat is Normal-form game, Nash equilibrium, mixed strategy, zero-sum game?

A normal-form game is a set of players, a set of strategies for each and a payoff to each player for every profile of strategies. A mixed strategy is a probability distribution over a player’s strategies. A Nash equilibrium is a profile of (mixed) strategies in which no player gains by deviating alone. A zero-sum game is a two-player game whose payoffs sum to zero; with a payoff matrix $A$ for the row player, mixed strategies $x$ and $y$ give her $x^\top Ay$.