What is a zero-sum game?

A zero-sum game is a competition/conflict in which 'the winner takes all'. Whatever is gained on one side is lost on the other.

Where does the term come from?

The 'zero-sum' is a mathematical reference to the fact that no additional value is added - one side must take from the other.

This video summarises the theory:

What is a typical zero sum game?

Zero-sum games are most common when there is a binary choice: a World Cup Final, for example, or a two candidate US presidential election. One side loses at the expense of the other and there is no shared interest.  

Isn't every contest like this?

In some competitions there there can be secondary gains that emerge from defeat. An example is the current Republican primary process in which the party is choosing its nominee for the 2016 Presidential Election in November.  
Here the calculation about the benefits of winning and losing may be different. While all candidates want to win, they may also have an interest in who loses (if candidate A comes second that might benefit candidate D but not candidate E etc.)

They may also be calculating the likely personal benefit of another candidate's ultimate victory - being chosen as candidate for Vice President, for example.

Why is it important to calculate whether a contest is really zero sum?

The EU referendum in the UK exemplifies the zero-sum concept. This can be difficult for some politicians to adjust to: 
“It’s a yes or no choice, Jeremy, in or out. There is no ‘maybe’ or ‘Only if the EU does what I want’ on the ballot paper.”  source