The Corleones are both a family and a "family"—one of New York's five mafia families. They're pretty tight-knit: "Don't ever take sides with anyone against the family again," says Michael when his brother Fredo gets out of line.
Loyalty to the family is a more important principle for most of The Godfather's characters than loyalty to any other principle or institution, like religion or the nation's laws. For a lot of people, family is a basic part of the order of society. But for the Corleones, it's actually an alternative to that order—it's a way of working outside the system to get the things that you want.
Nothing is more important than family.
Some things are, actually, more important than family.
Criminality is a super-obvious theme for The Godfather. The whole thing is about a criminal enterprise and the problems that arise from running it. The Corleones aren't anarchistic criminals—they're not randomly torching buildings and poking peoples eyes out with pencils (like The Joker in The Dark Knight).
They're criminals with their own idea of order. Don Corleone objects to dealing drugs because it's a "dirty business"—and he doesn't want to compromise his political connections. He treats crime like a business, and he has ideas about more acceptable and less acceptable business practices. Killing a horse and cutting its head off to terrorize a movie producer? Acceptable. Making a dope deal with the unscrupulous Sollozzo? Not so much.
"Thieves respect property. They merely wish the property to become their property that they may more perfectly respect it."—G.K. Chesterton
"Behind every great fortune there is a crime."—Honor de Balzac (the epigraph to Mario Puzo's novel, The Godfather)
"There's no honor among thieves" is a well-known proverb. But, in The Godfather's world, there's at least a little honor among thieves. Don Corleone's attitude toward drugs is a fair example (even if he's mainly doing it so he won't lose his political connections).
For the most part though, these criminals are pretty deceitful. Michael's bodyguard, Fabrizio, tries to assassinate him, murdering Michael's wife instead. The Corleone family's henchman, Tessio, turns traitor and sides with their rival, Don Barzini. Carlo, Connie's husband, helps lead Sonny into a death trap. Lies and deceit are everywhere in this movie.
"The only form of lying that is absolutely beyond reproach is lying for its own sake."—Oscar Wilde
"Well, when I was younger, I lied all the time, because once you understand the power of lying, it's really like magic because you transform reality for people."—Louis C.K.
"Revenge is a dish best served cold": Michael doesn't always follow that cliché, but at the end he does. When he shoots McCluskey and Sollozzo, he's not really serving it cold. But when he slaughters all of his and the family's enemies at the end, totally cleaning house, he's really living up to the cliché.
Usually, movies about revenge make revenge look like a pretty messy process: It doesn't work out the way you'd expect or hope. In this case, though, we don't really get to see the aftermath of Michael's final massacre—The Godfather Part II will show the possibly messy implications of all that. But we do see how revenge and violence are cyclical. For example, Carlo beats his wife, which leads Sonny to beat him up, which leads Carlo to help plot Sonny's death, which leads Michael to murder Carlo at the end.
"The best revenge is massive success."—Frank Sinatra
"The best revenge is to be unlike him who performed the injury."—Marcus Aurelius
The Godfather's Corleone family is in a weird position in America. Technically, they're rich and they enjoy lots of political influence. But they're also perceived as interlopers, as a threatening and alien presence, since they make their living through crime.
In order to make their way in America, they refuse to climb the social ladder in a legal way, because they feel like the system isn't really going to work in their favor. So they manage to achieve status for themselves—but it's not the best kind of status, given that it's marked by tons o' bloodshed. Vito wanted Michael to be able to attain a more legit version of status, becoming an honest senator or governor, but the business he (Vito) has created ends up sucking his son in.
"The rich are different from you and I"—F. Scott Fitzgerald
"Yes, they have more money"—Ernest Hemingway (responding to Fitzgerald)
The Godfather's version of the Mafia is an old-school, European, semi-feudal organization. The Corleones are in America, the New World, but they're relying on Old World methods: They don't use American democracy or the legal system to solve their problems.
Instead, they solve 'em through force and by relying on ancient notions of familial loyalty and blood. The role that Vito and Michael each take as the literal "Godfather" to family members symbolizes this: They're people you can view as a substitute father/president/judge, who can help you solve your problems if you agree to owe them loyalty and favors in the future. This system is an I-scratch-your-back-you-scratch-mine kind of thing.
"It takes an endless amount of history to make even a little tradition."—Henry James
"We don't want tradition. We want to live in the present, and the only history that is worth a tinker's dam is the history we make today."—Henry Ford