The best way to describe zero-sum mentality is by example. For instance, I bet you US$5 on a coin flip. There will be a winner who wins $5 and one person who loses it (i.e. give it to the winner). No additional "value" is created (or destroyed), so a coin toss is considered a zero-sum game.
In a more general sense, the zero-sum mindset is that there must be a winner and a loser in a given situation.
Wikipedia has a more mathematical description if you so wish to understand it better.