A Non-Zero-Sum Game is a situation where one's win does not necessarily mean another's loss, and one's loss does not necessarily mean that the other party wins. In a Non-Zero-Sum Game, all parties could gain, or all parties could lose. This is in direct contrast to a Zero-Sum Game where one party's win necessitates another party's loss, such as in competitive games like basketball, where if one team wins, the other automatically loses.
A classic example of a Non-Zero-Sum Game situation is called the Prisoner's Dilemma, where two prisoners are interrogated separately, and are offered a bargain where if one confesses, he is set free, while the other prisoner is convicted for 10 years. If both confess, they both face 2 years in prison. If both keep their mouth shut, they would both serve 6 months for a minor crime. Obviously, the optimal choice for both of them would be to keep silent and serve the 6 months. But since they don't know what the other person intends to do, they will both fear that the other one might confess, leaving them to suffer the 10 years. This will then pressure them into protecting what they see as their best interest, and so they will tend to confess, resulting in both of them suffering a loss, and being convicted for 2 years.