A zero sum game is one where whatever one person gains another person loses.
It is called a zero-sum game because each gain by one player, say +$100, is offset by the loss of another, -$100. The two always add up to zero.
A non-zero sum game is one in which the total amount of stuff that the players can win between them goes up or down.
Let’s take a poker game as an example:
Let’s say that there are three players, Dan, Bill and Dave each starting with $100, a total of $300.
They meet at Dan’s house and play for a couple of hours. At the end of the evening Dan has $200, Bill has $60 and Dave has $40. The total amount of money between them is still $300.
Dan is up $100, Bill is down $40 and Dave is down $60.
The total of these three numbers is zero (100-40-60), so it is a zero-sum game.
The next week they decided to head down to their local casino to play in a tournament. There are 50 players in the tournament, each paying $100 to enter, a total of $5,000. The casino takes a 5% cut, or $250, leaving $4,750 as prize-money for the players. The players start with $5,000 but they end up with $4,750, the total of their gains and losses is -$250 so this is a non-zero sum game.