The Godfather/YMMV

Adaptation Displacement: The original novel by Mario Puzo is less well-known than the films. The way that the films are titled "Mario Puzo's The Godfather" were actually an attempt by director Francis Ford Coppola to avoid this.
Alas, Poor Villain: Micheal Corleone. Even though he probably deserves every last bit, it's hard not to feel sorry for the guy.
Non Sequitur Scene: There was a quite lengthy subplot in the original novel about one of Sonny's mistresses who went out to Hollywood, became friends with Johnny Fontane, and eventually fell in love with a plastic surgeon who performed reconstructive surgery on her vagina and then married her. Francis Ford Coppola later said he was so disturbed by this portion of the book that it almost put him off filming it. Part III ignores this entirely by introducing Vincent, who is the same mistress's son (the book makes a point of saying Sonny never knocked her up before he was killed).
Broken Ace: Michael in Part II, especially towards the end. He is the most powerful Mafia Don in the country, has secured the Corleone Family's power and prosperity and eliminated all his enemies but he has alienated those who love him and relinquished his own happiness in the process.
Complete Monster: Though his role in the film is quite small, Luca Brasi in the novel qualifies as this.
Contested Sequel: The Godfather Part III. Opinions range from "absolutely terrible" to "not terrible, but not as good in comparison to the two masterpieces that preceded it". Some people contend that Part III would be a great movie, if not for the comically bad performance of Sophia Coppola as Michael's daughter. It would have also been nice to have Robert Duvall back as Tom Hagen.
Crowning Music of Awesome: The Theme.
Ensemble Darkhorse: Luca Brasi is only in the first film for a few minutes, and has less than five minutes of screentime. He's also very popular, and in the first video game, was featured prominently as a sort of mentor.
Even Better Sequel: This series has what most critics consider to be the Most Triumphant Example; Part II vs. the original.
Fanon Discontinuity: The first and second films are regarded as some of the best films ever created. The third one on the other hand... see Contested Sequel above.
Fight Scene Failure: When Sonny beats down Carlo, one of his punches very obviously misses by a mile, but comes with an impact sound effect and reaction. Ironically, James Caan really did injure Gianni Russo quite a lot during the filming of this scene.
Fridge Brilliance: In Part I, Vito says to the other dons, "I swear, on the souls of my grandchildren, that I will not be the one to break the peace we've made here today." Technically he kept his promise, but in Part III his granddaughter Mary, an innocent civilian, is killed by a bullet meant for Michael.
Genius Bonus: The establishing shot of the meeting between Vito and the heads of the other five families shows that it is being held in the Federal Reserve.
Harsher in Hindsight, or Hilarious in Hindsight: The Vatican plot of Part III, now that we know all the shady business deals involving the Papacy depicted in the movie doesn't even come close to the horrible things the Catholic Church has covered up in real life!
It Was His Sled: It is basically impossible to watch this movie without already knowing some of the major plot twists, due to Popcultural Osmosis and Memetic Mutation. In fact, if you are on this website, reading these words, it' is already too late.
Magnificent Bastard: Vito. Michael flirts with it but lacks the raw charisma to truly clinch the deal.
Memetic Badass: Arguably the whole movie is a Memetic Badass.
Memetic Mutation: "An offer he can't refuse", the horse's head, "the day of my daughter's wedding", "may your first foist child be a masculine child", the first known use of the phrase "badabing!", and many more.
Misaimed Fandom: More than a few people have, when discussing the film, referred to Michael as the ultimate Bad Ass. Smart, powerful, decisive, etc. That is not the point of his character arc. His story is a tragedy. Real-life mobsters are huge fans of the trilogy.
Moral Event Horizon: It's kinda hard to place one in a movie where The Mafia is involved, but what happens to that poor horse in Part I is unforgettably horrific... and it's done by the respective good guys. It's almost forgivable because the horse's owner - Woltz - is revealed as a pedophile using his position as film producer to seduce young women.
- Part II's Moral Event Horizon is much clearer where Michael has his own brother Fredo killed for unwittingly betraying Michael to Hyman Roth. Even when it's become obvious that there was no need for the killing. It's so painful that Part III can be viewed as Michael's attempt to atone for the sin of fratricide by seeking salvation allying with the Catholic Church.
  - Fredo though, at least had done something to deserve it. Arguably worse, and the starkest illustration of the moral emptiness at the heart of Michael's pretensions of family, is the casual murder of the prostitute to frame Senator Geary. She just happened to be the girl with him that night and her death was good for business.
Never Live It Down: The notion that Frank Sinatra used the mob to get his role in From Here to Eternity (that he got his Academy Award for) is not based on any evidence from Real Life, but this book and movie.
Nightmare Fuel: The scene in which Woltz, after actually refusing an offer he couldn't refuse, wakes up in bed the next morning with the bloody head of his favorite horse. Creepier still is that they used the head of a real dead horse.
Paranoia Fuel: Woltz has this realisation in the novel after that (in)famous scene, recognizing that if The Mafia could sneak onto his grounds and do the deed they did... In the sequel novels written by Mark Winegardner, when Tom Hagen is obligated to pay Woltz a visit several years later, he finds that much of the tasteful statuary and landscaping on the property have been removed and replaced in favor of greater visibility and security, giving the house a stark, almost fortress-like feel.
Tear Jerker: Toward the end of Part I, after Vito and Michael have had a final consultation, Vito is hit by an overwhelming wave of sadness, as he looks back on all the things he's done in his life. He says he Did What He Had To Do, and he has no regrets. Well, except one: that it's Michael who must now carry on the family business. He confesses that he once had hopes that the Corleones could be legitimately powerful, maybe even have had a mayor or a senator in the family...and that he had hopes that Michael could have been that. The broken way he says to Michael, "I never wanted this for you," speaks volumes.
The Scrappy: Mary Corleone. Sofia Coppola's performance as Mary in the third film is hated by nearly all. Compounding the issue is the apparent nepotism of her casting. She was actually cast only because Winona Ryder backed out at the last minute and there was no time to get another actress. Sofia fared MUCH better as a film director.
The Woobie: Fredo. Oh, Fredo.

For the games:

Anticlimax Boss - While bosses may wear body armour and wield heavy firepower, a single Boom! Headshot! will end them every time.
Demonic Spiders - Damn Tommy gun and shotgun mobsters will tear you a new one for a lot of the game if you're not careful.
It's Easy, So It Sucks - A charge often leveled at the sequel. Players often didn't bother with the "hunting the rival family's made men" sidequests because the benefit was negligible when it came time to take down the family compound. Elite Mooks, up to and almost including the family Don, were almost indistinguishable from ordinary buttonmen.
No Problem With Licensed Games: Neither will win awards, but they are still solid and fun sandbox games.
Player Punch - It seems like nearly everyone you know ends up getting killed by other mobsters, or betraying you and then getting killed by you. Luca Brasi, Paulie Gatto, Frankie, Sonny, Monk, Tessio, and Jaggy Jovino (and Sergeant Ferreira in the Wii version). YMMV because you never see these characters outside of missions anyway, so it's hard to get attached to them. Still, you gotta give the game some credit for trying.