From SA Mafia Wiki
A Lyncher is a third party role whose win condition is to lynch in the thread - the Lynchee. The Lyncher's win condition is not met if the Lynchee is killed in any way besides a lynch, such as from a nightkill. Typically, if and when the condition is achieved, the Lyncher is removed from the game.
Since Lynchees are almost always townies, they can be seen as scum, since their win condition would thus contradict the town's and advance the scum's by wasting a day's lynch. Generally the Lynchee does not know they are a Lynchee.
Winning as a Lyncher depends strongly on the role ability of the target, the relative ability of the other players, and the speed of the game. When designing games, it is generally a bad idea to have a Lyncher target a power role such as Cop (who can claim and keep himself from being lynched) or scum targets (as that makes the Lyncher less of a "third party" and more of a "weird townie").
Occasionally Lynchers have more than one target to get lynched, or they have to lynch a certain number of a certain group of players (e.g., "lynch 50% of power roles, including mafia"). However, as should be expected, more players to lynch leads to a harder game for the lyncher.
Lynchers should not be confused with the setup of the same name, which generally consists of 5 players: 1 lyncher, 1 mafia, and 3 townies (one of which is a Lynchee). The town wins when it lynches the mafia, the mafia wins when everyone else is dead, and the lyncher wins when the Lynchee is lynched. The game is over as soon as one team wins.
Lynchers are the "opposite" of Unlynchers.