Paramount and Alfran Productions tried to hook twelve other directors onto the project before finally asking Francis Ford Coppola. That's right: twelve. And Coppola, being a relative no-name, was under the threat of being fired constantly. Oh, and he was drowning in debt, owing Warner Bros. hundreds of thousands of dollars. It was an intense time, but the dude made it through.
Coppola initially fancied himself a bit of an art-house director: He wanted to do weird, personal movies. And he did go on to do that stuff later—along with other major masterpieces like The Godfather Part II and blockbusters like Bram Stoker's Dracula.
But before he'd made his mark, he had to labor intensively on the first Godfather movie and run the risk of losing his job multiple times. It paid off. Plenty of people will argue that it's still the best thing he ever did. (It's #2 on the American Film Institute's list of the all-time greatest movies, right after Citizen Kane.)
Coppola's decisions were always questioned by Paramount. He wanted an unknown actor by the name of Al Pacino to play Michael, but Paramount thought he was too short. They suggested that Warren Beatty play the role, at one point. But Coppola got his way. He even managed to cast some of his family members in the movie—his sister, Talia Shire, plays Connie, and his infant daughter plays the baby being baptized at the end of the movie. (That same infant daughter, Sofia Coppola, would grow up to write and direct Lost in Translation.)
If you want to read more about Coppola's difficult experience on The Godfather's set, check out this article from Vanity Fair.
Also, the dude knows how to do lighting. He has a painterly eye. This was something he had to fight for too: The producers thought he was filming the scenes in too much darkness, when he was really just setting a new, atmospheric standard for the way lighting would work.
Two maverick bad boys (actually, they were nice, normal people), Mario Puzo and Francis Ford Coppola, wrote the screenplay, based off Puzo's smash success of a novel (originally entitled Mafia before he changed it to The Godfather). But the road to the eventual movie masterpiece was a long and winding one.
Puzo wrote his book based partly off conversations with actual mobsters in Las Vegas—though he himself was never mafia-connected (granted, he did dig himself out of massive gambling debts by writing the book). The novel became a hit, and Paramount was all over it. Finally, after being signed to direct, Coppola began adapting and re-writing the script with Puzo.
Initially, however, Coppola wasn't a huge fan of the book itself: He didn't like the gratuitous sex scenes. But, despite these reservations, he ended up agreeing to team with Puzo. (Puzo would later go on to write the script for Superman II, commonly considered the best of the Superman movies, by the way.) Although he hadn't vibed with Puzo's book, Coppola found that he actually liked Puzo the man—they ended up working well together. They were able to craft a vision of organized crime in America and its strange and illicit re-working of the American Dream.
For more, check out this article in Vanity Fair.
Paramount Pictures and Alfran Productions gambled a lot on The Godfather. Paramount wasn't doing too well at the time, and the producers—despite hounding Coppola throughout the process—were actually letting a lot ride on the young, untested director. In the end, it was one of the most successful gambles in cinematic history, and the men responsible were Al Ruddy of Alfran and Robert Evans and Peter Bart of Paramount.
Originally, Paramount didn't want to spend the dough to film the movie in New York City and Sicily—the places where it was set in the book. They wanted to film it for cheap in Kansas City and back in their studios in California, and to change the setting to the present day. But Coppola had other ideas—he made sure that it was filmed on location and set in the 1940s.
The Mafia, however, were not grateful for the attention. The head of the Colombo crime family in New York actually raised money to try to prevent the movie from being made. He claimed that he wasn't really a gangster, and movies like The Godfather just perpetuated negative stereotypes (which is a pretty clever way of trying to protect your criminal enterprise, admittedly). Frank Sinatra opposed the movie, too, since the character of Johnny Fontane was obviously based on him. But it happened, after many tribulations—all detailed in this article from Vanity Fair.
The Godfather was shot on 35 mm film—kickin' it old-school (because it really was old school and digital film hadn't been invented yet). Additionally, it was filmed on location in New York City and Sicily, although Coppola had to fight for this to happen. The producers initially wanted to shoot it in Kansas City or on their own studio's lot.
One cool fact about the film is that the cinematographer, Gordon Willis, insisted on shooting every scene from angles mimicking a first-person, human perspective. There's just one major deviation from this: Coppola convinced Willis to shoot part of the scene where Don Corleone is almost assassinated from an aerial angle, telling him that this represented God's perspective on the event.
Coppola also used groundbreaking lighting in the movie, insisting on using shadows and darkness to create effect. His more painterly attitude toward the mixture of light and shadow became very influential.
Who doesn't want to hear this classic every time their cell rings? The main theme to The Godfather has become legendary, referenced in various parodies and tributes.
There's something strangely moving and evocative about that trumpet. It's hard to even say what feeling it expresses—whether it's sadness, some kind of longing, or a certain mystery or curiosity. There's something vaguely romantic about it, bringing back to mind the Old Country, Sicily. Whatever it signifies, it sticks with you.
We hear it right at the beginning, as the Paramount Logo and The Godfather "Puppet-Master" title-card come up. Yet it drops off and we're confronted with the movie's famous opening line, "I believe in America."
The music almost functions as a calling card for the Don, for the idea of this older Sicilian order. Bonasera, angered and aggrieved at the brutal attack his daughter suffered, admits that his belief in America has been tarnished, and now, he's turning to Don Corleone for help. The music seems to signal that older, Sicilian way of doing things, that secret order.
The man responsible for this soundtrack was Nino Rota, an Italian composer. He actually won an academy award—not for this movie's soundtrack, but for the soundtrack to The Godfather Part II.
He also composed the score for some of the great Italian director Federico Fellini's most famous films, like La Dolce Vita, La Strada, 8 ½, and Amarcord, along with Franco Zeffireli's Romeo and Juliet (frequently shown in high school English classes).
Tons of people love the Mafia (or think that they love the mafia—they love their idea of it, at any rate. In reality, it's a murderous, wholly negative organization). It has to be the most successful criminal organization in its ability to inspire fictionalized romance, despite its vicious reality. (Maybe The Departed and The Town helped beef the Boston mob up in that area, as well?)
For instance, few people are cultish followers of the Russian Mafia—Eastern Promises made that organization look fairly bleak, unless you don't consider naked knife fights at spas to be brutal and horrifying. But the Italian-American Mafia has the panache the customs to momentarily delude people into believing that they want to be in it.
The Godfather is the central cultural artifact in this obsession. Sure, there's Goodfellas and The Sopranos… and those are both important. But The Godfather really gave life to the whole construction of the mythology of mob life. This displeases a lot of people, although that's not necessarily The Godfather's fault. Nevertheless, the mafia's mythos soldiers on.
Why is this the case? Maybe it's because the mob, with its five storied families in New York, with its hierarchy, with its customs and codes, with its violent anecdotes and legends, has all the detail and mythology of Game of Thrones or Star Wars. At this point, thanks to the culture, it has its own universe, existing in both fictitious and real versions.
As with people who've memorized all the moons of Naboo and Endor and who know all the heirs to the House of Lannister, people who get deep into the Mafia's actual and fictitious history are probably doing it more for the fascination of the mythos—to lose themselves in an imagined world where the normal rules don't apply—than because they're aspiring to become actual criminals. It's a huge subculture.
You can see the evidence of that subculture and its relation to The Godfather all over the internet. Check out the Godfather Wiki for a good example. It gives you an in-depth survey of the mythology of the three movies, along with background from the books, conveying a sense of the vast scope of the world that Puzo, Coppola, and others managed to create.