The Gotham mafia may be one of my favorite Batman villains. No one else seems to fawn over them the way fans do for any of the other Batman villains. Looking up Falcone or Maroni fan art on the internet is useless. I can find a billion awful Harley Quinn pieces, but Carmine Falcone turns up just a few scans from comic pages.
Batman's origin always included becoming Batman to fight crime in Gotham, a city with a high crime rate. Miller, I think, wanted to ground his Batman in reality, and historically big cities had organized crime problems. Gotham, being the culmination of every corrupted, dark, and dangerous part of every city everywhere, should have organized crime.
Bob Kane and Bill Finger created Sal Maroni as a mobster responsible for disfiguring Two Face, and created Tony Zucco as the mobster who put a hit on Dick Grayson's parents. But organized crime didn't take a front seat to Batman stories until 1987. Frank Miller rebuilt the Batman mythos, taking it back to the 1940s stories of Kane and Finger, which meant darker, grittier stories. The Batman:Year One origin debuted Gotham's organized crime family, the Falcones.
Jeph Loeb's The Long Halloween expanded the Falcone Crime Family, basing it off of the Corleones from the Godfather. He tied the Maronis and Tony Zucco into the story as well, but maintained that the Roman Empire, as Carmine "the Roman" Falcone's criminal organization is known, had the most power and influence.
The pulp theme of mafia families holding so much power that only a vigilante operating outside the law can take them down can be seen across all entertainment mediums, in novels, comics, and movies. The Long Halloween/Dark Victory storyline helps transition Gotham organized crime from realistic mobsters to comicbook villains like the Black Mask and Maxie Zeus by showing the downfall in a who-dun-it serial killer murder mystery. Spoilers follow, just so you know...
The head of the Falcone crime family. Took power from his father, Vincent, the founder of the mob family. His sister Carla Vitti lives in Chicago and is the head of a crime family there. Known as the Roman. Created by Frank Miller in 1987. He has claw marks across his face, courtesy of Catwoman. He is murdered by Two Face.
Changes her name to Sofia Gigante, also known as the Hangman. She poses as crippled in a wheelchair, and takes over the family business after her father Carmine dies. Gigante murders ten people connected with Harvey Dent and his rise to District Attorney. She is also killed by Two Face.
Based on Michael Corleone, Mario is the elder son of Carmine. He is absent for most of the Long Halloween and Dark Victory storylines, as he has been living in Italy undercover, much like Michael Corleone in the Godfather. When Mario returns to find his family mostly dead and the business in ruins, he scuttles the rest, burns the house down and rebuilds.
The younger son of Carmine, Alberto is also the Holiday Killer. He fakes his own death and starts targeting Maroni Family members, killing a dozen including Salvatore and Luigi Maroni, and his own aunt Carla Vitti. He is later killed by his sister Sofia.
Created by Bob Kane and Bill Finger in 1940 as a vehicle for creating Two Face, Sal Maroni is the mob boss responsible for scarring Harvey Dent with acid. He is the son of Luigi Maroni, and father of CJ, Pino, and Umberto Maroni. The Maronis and Falcones have a long history of rivalry and competition. Salvatore is killed by Alberto Falcone.
Also known as Boss Zucco, he was created by Bob Kane and Bill Finger also in 1940, and also as a vehicle to introduce another more important character. Zucco is the mobster responsible for extorting the circus, and sabotaging the high wire act, killing Dick Grayson's parents. Jeph Loeb included Zucco as an underboss in the Maroni crime family. Dick Grayson and Batman scare Zucco so much, he has a heart attack.