Answer by Carrie Rickey:
A big question, but I'll offer a thumbnail.
The gangster film is one of the most enduring of Hollywood genres. In America the movies have mythologized three types of heroes: the cowboy, the cop, and the gangster. Guys with guns in a world without women. Most durable is the gangster, for without him, the cop has no adversary.
And when he’s also a modern cowboy whose frontier is the city – or, as with Michael Corleone in “The Godfather Part III,” the world – the gangster gives us hero and villain in the same character. That’s why the taste for gangsters has abated only briefly – during World Wars I and II, when real armed conflicts sated any pangs for reel mob wars.
"The Godfather" saga has so much resonance because it taps into both the gangster myth as well as the immigrant myth. At it heart it's a story of a single family, refugees of feudal conflict, and their assimilation and success in the New World where, unconsciously perhaps, they replicate the feudal structure from which they fled. Thus the multigenerational saga also has the contours of a multigenerational classical tragedy, like the Oresteia.
The Mario Puzo/Francis Coppola conception of the Corleones is at once a portrait of a specific family with its sibling rivalries and love as well as a portrait of America, a nation of immigrants struggling for power and legitimacy.
To underline its timeless tragedy,Coppola shot the film with the chiaroscuro light and shadow of classical painting.
And this story of the generational shift within an immigrant dynasty exemplified the generational shift in Hollywood by casting Marlon Brando, rebel of a prior generation, as the patriarch and introducing a new generation of actors, relative unknowns such as Al Pacino, James Caan, Robert Duvall and John Cazale.
Also, what a contemporary viewer might miss in The Godfather is its coded anti-Vietnam message to viewers who saw it in 1972: When Don Corleone is in the hospital, casualty of an internicine mob war that has taken his eldest son, , he states categorically that he doesn't want to see the younger generation die: "The war stops now!"
A million years ago I wrote a piece about the gangster movie that explores these themes at greater length: