A zero sum game is one in which the total pay-offs are the same for all possible combinations of players' strategies. This means that each player can only gain at the expense of others; any player's loss is balanced by an equal gain (or gains) made by other players.
It does not actually matter much whether the sum of the pay-offs is zero (the strict requirement for the zero sum game) or another number, as long as the total is the same for all possible outcomes. For this reason, zero sum games are often called constant sum games.
A real life example of a zero sum game is gambling. If one player wins, other players (including the house, if any) must have lost the same amount.
Financial markets offer many examples of zero sum games. For example the writer of an option can only gain what the holder loses and vice-versa. Although securities markets are expected to have an overall positive outcome, attempts to out-perform are zero-sum, in that all out-performance by winning investors must be balanced by out-performance by losers.