Not to repeat too much what's already been said, but the
mafia is just an italian word for their mob, usually sicilian. In sicily back in the
old days, these guys were like, well, warlords. You went to them for resolution conflict, you went to them for loans and capital investiment, you
went to them for laws and everything else.
A mob and a gang are just the more generalized, unspecific terms. So people who have been made in the mafia are mobsters and gangsters.
Consider, for example,
Mara Salvatrura, aka MS-13. A person who's been initiated into MS-13 is a Gangster, an MS-13er or whatever. But
Mara Salvatrusca is simply spanish for "The El Salvadorian Mob" So when you say someone's in the Mafia, your really just saying that that
person is a gangster. And in the strict technical sense, it really only refers to an actual organization of thugs and goons that came out of sicily
and were eventually setup as an Organized Criminal Syndicate in the US, ostensibly by Lucky Luciano. Anything 'outside' of that organization is not
the 'mafia'
sensu stricto, but, agian, mafia has simply become, today, a term for any organized italian criminal gang.
And of course there are the spin offs, because the Italian Mobsters were so infamous that their word for 'criminal family' became the standard.
Supposedly, if highlanders from scotland had started organized crime in teh US, we'd probably refer to them as The Clan, and talk about the Russian
Clan and the Persian Clan, etc etc. So the Russian Mafia wasn't organized by a suborganization of Lucky Luciano's Organized Mafia, etc etc.
Here's a sicilian page on it.