SCREEEEEEEEEEEEEEEEEEEEEEEEEEEE!!Explanation The screeching sound the animatronics make when they attack you has spawned various onomatopoeia, with this being the most commonly-used one.
What does the fox say? "SCREEEEEEEEEEEEEE!!"Explanation A reference to the already-memetic Ylvis song "The Fox" with Foxy's Jump Scare sound.
A duck walked up to a lemonade stand and he said to the man running the stand: "SCREEEEEEEEEEEEEE!!"Explanation In a similar stem to the meme above, it's a reference to "The Duck Song", associated with the Fandom-Enraging Misconception about Chica being seen as a duck rather than a chicken.
4/20Explanation A Fan Nickname for the custom level in which the player sets all robots' AI to 20 (most difficult). Jokes have been made due to the number in question.
Edits of various characters peering into the booth in the place of Foxy have taken off.
Jokes about Foxy's speed have become prominent, usually comparing him to Usain Bolt, Sonic the Hedgehog, or The Flash.
There's a comic, of which several variations have since been made, in which Foxy runs into the office, only to encounter something unexpected that scares him.
Perhaps due to the restaurant setting, gags featuring the animatronics acting out parts from SpongeBob SquarePants — particularly those set in The Krusty Krab — have also cropped up.
"I chimed in with a 'Haven't you people ever heard of... closing the goddamn door?!'." Explanation The beginning of the chorus to "I Write Sins Not Tragedies" by Panic! at the Disco has been memetically associated with the game, as you need to close the office doors to defend yourself against the animatronics.
Five Nights At Freddy's Troll Palacenote An animation by Crazy Boris Production depicting the animatronics just wanting to play Scrabble, and the player being someone who kept missing in pressing the correct button. It was only an animation, but somehow it is often remade using other characters from various media, and making people think that this could be Five Nights At Freddy's in a nutshell.
Pizza? Pizza! Pizzaaaaaa. PIZZAAAAAAAAAAAAAAA! Explanation "How to Make Five Nights at Freddy's Not Scary" and its follow-ups has, as a Running Gag, Chica only capable of saying 'bawk' and 'pizza' in various tones and inflections, which also became her running joke in Supermarioglitchy4's Super Mario 64 Bloopers. It became an Ascended Meme in Security Breach, with that game's Chica shouting 'pizza' three times before she tackles a S.T.A.F.F. bot carrying an order meant to distract her.
Foxy is a good guy. Explanation Some people, whether as a joke or a serious theory, speculated that Foxy was Good All Along on the grounds that his Jump Scare isn't as aggressive as the others'. The theory suggested that the reason you die is that you get a fatal heart attack from the sheer fright, making it more of an Accidental Murder. Of course, this was jossed in later installments but is still referenced.
"Uh, the animatronic characters here do get a bit quirky at night." Explanation FNAF's Establishing Series Moment is the first phone call of FNAF1, in which Phone Guy tries to explain to you the animatronics' nighttime behavior while downplaying the severity. A common joke is to juxtapose this quote with something either Faux Horrific or because of the vagueness of "quirky", just about anything. Examples include Freddy (usually Withered Freddy) twerking, Toy Chica in suggestive poses, and the animatronics having a dance party.
FOWA!Explanation The videos by EthGoesBoom have Bonnie pronounce four in a western accent as "fowa", which annoys Foxy to no end, and led to a prank war video of The Scary Maze Game where Bonnie spammed it relentlessly. The word has been used by many other videos, and received a Sparta Remix.
The infamous "DOOR STUCK" meme has been referenced various times when referring to the game mechanic of the doors suddenly jamming.