Lynch someone = "execute" - used to describe the crowd-based killing of one unlucky player every day in Mafia games.
In most Mafia games, the lynched player is the one who has the most votes at the end of the voting process (plurality). The lynched player is then eliminated from the game (becomes a ghost) and (usually) some information is revealed about their faction and/or role.
This mechanism is one of the defining features of the Mafia game, allowing the uninformed majority (Goodies) to have a way to eliminate the informed minority (Baddies or other factions) during the day cycle.
Because votes can be changed and sometimes at the end of a day, more players may have an equal number of votes cast on them, tie lynch rules are needed to determine which (if any) of the players is lynched.