The concept of the veto player is a political actor who has the ability to decline a choice being made. Specifically in Tsebelis' analysis a veto player is one who can stop a change from the status quo. This is analogous to players in a bargaining game where all players must reach agreement.
This is best seen in example. We can imagine that 3 flatmates, one of whom wants to buy a cat (person A). If person A buys the cat, and the others accept it, then there is one veto player person A. What person A chooses is the social choice. However it may be the case that the others can call animal control and have the cat removed. In this case, they are all veto players. If anyone chooses not to have a cat, then the social choice is not to have a cat.
A key feature of veto players is that they have preferences over public policy outcomes and these are continuous across the continuous policy choices the veto player faces.
There are a number of difficulties with applying the concept of veto players to political systems:
Having established the concept of veto players, Tsebelis the applies this to social choice, following Anthony Downs' approach of continuous policy space with veto players concerned solely about proximilty of choices to their ideal on a policy spectrum. Further he assumes that there is a status quo point (apparently analogous to a disagreement point in game theoretic bargaining analysis).
He argues that the status quo will only change if it is weakly preferred by all veto players (since otherwise one of the players would veto the social choice). This is analogous to saying that the status quo will only change if the status quo is not pareto efficient.
Tsebelis then suggests that where there is pareto improvements available, the social choice will be for a point which is Pareto efficient. He suggests that in the case where there are many such points, there will be mechanisms to determine which point is reached (although there is no explicit exposition of a bargaining analysis either co-operative or non-cooperative).
Tsebslis then looks at how various veto players resolve certain situation (changing the number of policy dimesions, veto players and status quo points). In so doing he looks at situations with many solutions.