A mobster is a bad guy who's involved in organized crime or belongs to a gang. There are many movies about mobsters that take place in the 1920s and 30s.
As opposed to individual criminals, members of crime syndicates and gangs belong to a large, structured criminal organization. Someone who belongs to such a group is a mobster or a gangster. The term mobster comes from mob, another name for the Mafia, an Italian organized crime group. Originally mob meant just "a large group of people" or "the common people," from the Latin phrase mobile vulgus, "fickle common people."