A play-off is a set of one or more matches to determine a winner among a group of players or teams. It can be used as a tie breaker in tournaments.
Play-offs are considered a very fair way to break ties, as they measure the performance of the tied players very directly. In most tournaments, however, there is usually not enough time to have a play-off, and other tie breakers are thus far more popular.
Play-offs are thus most often used in very important tournaments, such as major titles or national championships. For a list of examples of tournaments that use play-offs, see Play-off / Tournament Usage
A tournament can result of more players tieing for the same place. play off can be used in this case by organising a mini round robin or knock out tournament between these players. Unfortunedly a round robin tournament can result in another tie.
Avoiding another tie
One way to prevent another round of ties is to have a seeding (random, first to end, or otherwise) for the players. Then, for example in a 3-way tie, you can have the two lowest seeded players play each other, and then the winner of this game gets to play against the top seeded one.
Direct comparison is not a play off
Consider the following example:
Four players (named Adam, Bengt, Cthulhu and Donald) who play a round-robin of 3 rounds. Adam and Bengt get 2 wins each, and Cthulhu and Donald get one win each, like this:
A B C D Total Adam X 1 1 0 2 Bengt 0 X 1 1 2 Cthulhu 0 0 X 1 1 Donald 1 0 0 X 1
Now one might think that since Adam has won against Bengt, Adam must be the stronger player. However, it could also be argued that Bengt has won against Donald, who in turn has beaten Adam, and therefore Bengt is stronger player. Therefore any decision made on the basis of the direct comparison is likely to be controversial (see /discussion?).
A play off between Adam and Bengt, on the other hand, will give a whole game worth of extra data to make decisions by, so there should be more confidence in the final result.
Similar reasoning will also apply to any other situation where the tied players have the same SOS.