A zero-sum game is one where the gains by one player or participant are exactly even to the losses of the other player. An example would be two countries fighting over a piece of land. If one country gains it, then the other country loses it.
The phrase "zero-sum" can be best understood by looking at it from a mathematical point of view. For example, two people make a bet for $5. The person that wins now has positive $5, and the person that loses has negative $5. Adding +5 and -5 gives a sum of 0.
A win-win situation is an example that is not zero-sum. In this kind of situation, both participants gain something, so this is considered a positive-sum game.