A meme is something that spreads rapidly, becoming well known, usually around the Internet. The Simpsons has spawned a couple of memes over the years. An idiom is something that originated from something like a TV show, book, movie etc. but has became commonly used.
"That's a paddlin'"
Dead Bart is a story on Creepypasta that has become well known over the Internet. Many fans have been led to believe that it is real, although it is not.
The goggles do nothing!
Originating from the quote by Rainier Wolfcastle "My eyes! The goggles do nothing!" from the episode "Radioactive Man", the quote has been shortened to simply "the goggles do nothing". It is now a popular catchphrase used for conveying the horror of having seen something unwanted.
"Milhouse is Not a Meme"
The phrase "Milhouse is Not a Meme" a paradoxical statement and a well-known debate on 4chan about what makes an Internet meme and what doesn’t, using The Simpsons character Milhouse Van Houten as an example. Since the birth of the debate on 4chan in 2005, the phrase has been typically used to initiate a chain post of recursive nature. Due to its recurrence over time, “Milhouse is not a meme” is often referred to as a forced meme.
Originally from Skinner's story from the episode "22 Short Films About Springfield", this meme features Seymour Skinner, Gary Chalmers, and the "steamed hams" Skinner bought. Since 2015, this meme has gained big popularity, and has many characteristics, including Skinner's weird expressions (most notably the one he makes at the end of his story and the one he makes right after he says "Oh, no, I said steamed hams".), the phrase "Aurora Borealis", Gary Chalmers, and the actual steamed hams. These characteristics had been used by many Simpsons fans, including in photoshopped edits of Simpsons scenes, non-photoshop memes, etc.
Originally from the episode "Little Orphan Millie", this meme features a funny screen cap of Marge Simpson while krumping to make Bart better. Since 2016, this screen cap has gained big popularity, being used in Photoshopped photos of other Simpsons scenes, non-Simpsons photos, and in Photoshopped gifs.
Originally from "Lemon of Troy" and "Homer Badman", this meme actually features two tarty-expressions from Shelby's Father and Homer, respectively. Mostly, this meme features Shelby's Father's expression when he ate the lemon, with watery eyes and a tightly closed sour mouth. Sometimes, this meme features Homer's expression when he ate the 77X42, which looks the same as Shelby's Father's expression, but without watery eyes and has big wide eyes instead, with the same tightly closed sour mouth. Since 2016, this meme has gained popularity, being used by Simpsons fans on many other characters from the show besides Shelby's Father and Homer.
"Just Give 'Em One of These"
Originally from the episode "Secrets of a Successful Marriage", this meme features Moe Szyslak teaching his "Funk Dancing for Self-Defense" class, accompanied by his dancing. Since the summer of 2016, this meme has gained huge popularity on YouTube, being used in remix videos of him dancing to several songs, and also as a screen cap of Moe's first dance being used in photoshopped photos.
Cheese-eating surrender monkeys
"I, for one, welcome our new insect overlords"
While not originating from The Simpsons, meh has been thought to be popularized by the show.
D'oh has been added into the Oxford English Dictionary. It is commonly used by people both over the Internet and in real life.