It seems that if you are imaging a Mafiosa in your head, you're most likely picturing, if not a character from one of the Godfather films, certainly you have an image that is heavily influenced by the films. They were immediate hits, and few would disagree that at least two of them deserve every bit of praise received.
But while it is blazingly obvious how the films have influenced the image of the Mafia in America to us non-Made Men, what about for the people who it actually portrayed?
It is easy to see the figures as romanticized anti-heroes in many regards, so I would suspect that it was hard for the very subjects of the film, the Mafia itself, not to latch onto the films. Perhaps I'm being presumptuous in assuming that Puzo and Coppola took a fair bit of creative liberty in how they portray the Mafia in the film, but essentially, I'm wondering how much of the tail was wagging the dog in the wake of the film. Did the image of the Mafia in the films see the culture within the Mafia shift at all to reflect their media portrayal, and if so, how?
This question is a bit more tangled than it seems on the surface, because in the end both the real-life Mafia and the film drastically influenced each other. First, the author of the original book had never met a real life member of the Mafia before production of the movie began, neither had the film executives or Coppola himself. However, once the film was approved and production began there were "made-men" taking a very keen interest in the way they'd be indirectly portrayed.
In 1970, Joseph Colombo, real-life head of the Colombo Family in New York (one of the "Five Families"), actually launched a committee called the Italian-American Anti-Defamation League (later called the Italian-American Civil Rights League) to combat the poor reputation of Italian-Americans in the US. When production for the film began in 1971 the League approached producers and made a deal with them to have the film made but under supervision. As a result you never hear the words "Mafia" or "Cosa Nostra" in the first film. As a result of this there were real-life "mobsters" on set, and many of the actors swapped stories with, got tips from, and even directly mimicked the real thing.
Jumping forward, according to several sources including Diego Gambetta's Codes of the Underworld: How Criminal's Communicate, members of the Mafia have admitted to borrowing tactics from the films, certain signs of respect, and even direct quotes. Even the term "The Godfather" itself wasn't used within the Mafia until after the film released. So we can see that the movie has not only shifted the general culture of the Mafia, including the way they dress, act, celebrate, and even speak, but it's been useful for enforcing the hierarchy of the organization. Keeping so many men, many under-educated, in line when the line of work requires them to take violent action on a regular basis is a challenge, but the film has helped solidify and romanticize the way they do their business, and reinforces loyalty and the cost of stepping out of line.
Beyond that romanticism the Italian-American community as a whole was affected. The films featured gangsters who weren't just mindless criminals, they were fully fleshed out feeling, thinking human beings. This played a huge role in dragging Italians away from the stereotypes of the day and made them more human in eyes of bigots at a time when they were very much struggling for identity. This gave Italian-Americans both within and outside the Mafia a real identity, not just the cartoonish stereotype they'd been perceived as before, in both their own eyes and in the eyes of America in general.
So in the end while the Mafia itself heavily influenced the films, the films ended up influencing the Mafia as well.
Sources:
Birthmoviesdeath.com: The Time The Mafia Protested The Godfather
The Guardian: Why The Mafia Study Gangster Movies
Gambetta, D.: Codes of the Underworld: How Criminals Communicate
Santopietro, Tom.: The Godfather Effect: Changing Hollywood, America, and Me
Shortlist.com: The Mob vs The Godfather
SmithsonianMag.com: What is The Godfather Effect?
Edited to fix an inaccuracy in the use of the words "Mafia" and "Cosa Nostra".