Anvilicious: Dicks (hawks) are necessary because assholes (general threats to people) need to be fucked (dealt with) before they shit on (blow up) everything, but pussies (doves) can get full of shit too, becoming assholes. Pussies are necessary because without them, dicks "fuck too much (There Is No Kill Like Overkill) or fuck when it isn't appropriate (Disproportionate Retribution)." In short, extremism is bad, m'kay?
In political terms: "dicks" = conservatives; "pussies" = liberals; "assholes" = rogue states/extremists/terrorists.
Awesome Music: Harry Gregson-Williams deliberately scored the film completely straight, as if it was a real action movie, to help sell the satire. It works perfectly in the movie, but out of context, the score is genuinely exciting and rousing.
The film's stereotypes of foreigners are so ridiculously hilarious it's impossible to take them seriously.
Matt and Trey made the sex scene as dirty as possible, dirtier than they wanted it to be knowing the MPAA would want them to cut stuff out of it.
The scene where a very drunk Gary vomits outside the bar. Starts out at "small amount", then moves up to "disgusting but realistic", then graduates up to "large amounts", then "absurdly large amounts", then finally settling for "high pressure projectile vomiting at the camera and a puddle of the stuff at least 5 meters across". Don't forget the dramatic music.
The song "End of an Act" eviscerates Ben Affleck and praises Cuba Gooding Jr based on their work in Michael Bay's Pearl Harbor. The film was made in 2004, when Affleck was in the middle of his career slump and Gooding still living off the high of his Jerry Maguire Oscar win. Now, Affleck has made a significant comeback with his directorial works Gone Baby Gone and The Town (the latter of which also received praise for his performance) while Cuba is doing straight to DVD schlock after appearing in disasters like Norbit and Daddy Day Camp.
Also, Kim Jong-Il's death is finally starting to make sense, but the only difference is that it's just too bad that American dicks never fucked that asshole to begin with.
The main character's name is Gary Johnston. Eight years later, the Libertarian Presidential candidate was named Gary Johnson. Seeing how the filmmakers are known for their Libertarian viewpoints, they more than likely voted for a guy who almost had the exact same name as one of their own characters.
The "Dicks, pussies and assholes" speech is basically a more profane version of the "Sheep, sheepdogs and wolves" speech from American Sniper.
For that matter, Guardians of the Galaxy has a scene where Peter Quill seems to mix up the definition of "Dicks" and "A(ss)holes" as defined by this movie, saying that he and his team are "A-holes" but "not 100% dicks."
MATT DAMON! The real Matt Damon has said he gets this constantly.
English DJ Scott Mills actually adopted this into his show, and listeners who had never seen the film thought he came up with it. When Chris Moyles (who presents a different show on the same station) was discussing a Matt Damon movie shortly afterwards, they referenced the joke, and people texted in accusing him of stealing the joke from Mills. Moyles remarked that the joke was from Team America and pointed out that the film was several years old at that point.
I'm so ronery.
This has gone full-on to the point of becoming its own slang: The Engrish term became used to disparage extreme antisocial Otaku who look down on "real-world" relationships, and evolved to mock Perverse Sexual Lust among Periphery Demographics in general.
Many fans are in the US government and foreign policy field, and as explained in Misaimed Fandom below, many Americans love the theme song, "America, Fuck Yeah!".
There's even word that Kim Jong Il himself saw this film and was tickled by it, saying something along the lines of "This is a very funny movie." On the other hand, there's also word he wasn't impressed and tried to convince the Czech Republic to ban the film (for what it's worth, this is what Wikipedia has to say on the matter).
Hans Blix referenced his portrayal in the film (which he appears to have found as humorous as it was intended) in a recent lecture.
Most of the actors that were parodied loved it, and just about all of those actors stated for the record that they wish Parker and Stone had contacted them so they could voice themselvesacting like pussies.
Alec Baldwin cracked up when his daughter's classmates quoted Kim Jong Il's, "You are useress to me, Arec Bardwin!"
George Clooney said that he would have been insulted had he not been included in the film.
In fact, the only one of the film's targets who seems to have taken it personally is Sean Penn, who wrote Parker and Stone a very angry letter (telling them just how angry he was) after the film's release.
The entire "Dicks, Pussies and Assholes" speech works as an Intended Audience Reaction example of this. It should be utterly ridiculous but the emotion, the music, and surprisingly appropriate metaphors make it work so well, you'd be forgiven for forgetting Gary is just quoting the drunken ramblings of that bum from the bar.
Never Live It Down: Matt Damon's portrayal — being capable of saying only his own name. Though he found it funny (and wished he could have done the voice work), fans—especially one known as KSI—still bring it up at every chance.
Nightmare Fuel: Kim Jong Il's whole plan to create chaos and violence is deeply unsettling.
Uncanny Valley: Seriously, just look at the main page's image and you'll see. The faces are very unsettling. It may explain the directorial decision to leave the strings visible, to stop the marionettes looking too human.