A Jester is a role that wins when they are lynched. If they survive to endgame or are killed in any other way, they lose. Even though technically they might be lynched in the implicit process of scum endgaming a Town, they invariably just lose when the game gets to that point to avoid that kind of rules lawyering. Similarly, if the Jester is the only role in the game stopping the Town from winning, the Town simply wins and the Jester loses.
The original variant of Jester ended the game when it won (leading to things like BBB's Gambit). However, this almost always ended in short games where the Jester won. Thus, once the Jester is lynched the game usually continues. Jesters who are lynched are considered joint winners alongside whichever main faction wins the game.
Jester is held in profound distaste partly because it is easy to win with (any village idiot can get lynched Day 1) and partly because even in the best case it punishes Town for lynching someone who is not of their alignment.
Jesters have been seen with a few hindering modifiers attached to them, such as Even-Day (the Jester only wins if it is lynched on an even-numbered Day) or a post restriction. In other games, players are told that there is a Jester in the game. This does not usually change that the Jester eventually wins.
Based on all of the above, Jester is generally considered a bastard role.
This role is occasionally called, but has absolutely no relationship to, Lynchee.
Use and Power
Jester balances like a guaranteed mislynch. To balance this, sometimes when Jesters die the moderator enacts some pro-Town act to counterbalance this; for instance, the player who hammers a Jester wins a One-Shot Vig kill.
Most people will probably still tell you to simply not use Jesters though.