Director
Art House, or Blockbuster?
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 Own Family Business
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.