If "movie-accurate" is good enough for you, then read on.
Basically, organized crime is a protection racket, and everyone pays. This includes the criminals involved in organized crime. You join a family / crew / outfit / whatever to protect you from the other guys on the street. At the absolute lowest level, you have the legitimate wage-earners. These guys do the work who earn the families their money when their employers pay off the criminals. And if another outfit moves in and starts attacking these guys, you can have trouble.
Anyone who is actively working as a criminal in an organized crime outfit can go by a lot of names. "Associate" is common. These guys usually aren't strongly tied to the family, but they have to give the "made men" above them a cut of their earnings. If you've seen
Goodfellas, Ray Liotta, Robert De Niro, and Joe Pesci all play associates. Liotta and De Niro can't be "made men", since they're not 100% Sicilian. Pesci is trying to become a made man, but he isn't there for most of the film.
Above associates are the "made men", or "soldiers". These guys have proven their loyalty (and their potential as earners) and move up the pyramid. They get more territory to work with, and get to oversee some associates, recieving tribute from them.
The made men in turn kick up a percentage of their income to the "captains", who control even more territory. These are made men who have really proven themselves and earned consistently for years, providing their outfit with income, prestige, and power.
At the top of the heap, you have the boss, who takes his tribute from the captains. The boss controls the whole thing, with an underboss to help manage the day-to-day affairs (and to provide the boss some distance with the rest of the outfit for protection in the form of deniability in court) and an advisor or right-hand man ("Consigliere") who doesn't actually run anything, but helps the boss keep everything running smoothly. The boss heads the entire family, and with the underboss and the consigliere keeps everything running smoothly, managing the family's finances and resolving disputes. The boss is also the one who you're supposed to go to if you have a conflict with a member of his family. He's supposed to deal with the big picture and keep the peace (or win a war, depending on the times). These are the guys you hear called "Don" in
The Godfather, as a sign of respect.
Respect, by the way, is the reason why even retired members of a family are still given formal respect and given lavish titles like "the king boss of bosses" ("
capo di capi re" in the Sicily). Marlon Brando took on such a role as Vito Corleone after his retirement in the second half of
The Godfather. These guys aren't involved in the day-to-day anymore and really only get nominal tribute if any, usually in the form of the new boss taking care of them.
Regardless of the nationality or styling, organized crime pretty much works like this across the board. Wikipedia has a halfway decent overview of names in the Sicilians in America ("La Cossa Nostra")
here. You can do the same with the Mexican mafia, the Russian Mafia, the Irish Mafia, the Yakuza, South American drug cartels, etc. and be close enough for "movie accurate", just changing the names (which are linked to at the bottom of the above page, though I don't know enough about those organizations to vouch for accuracy).