Clip Art Animation

The cheapest way to create an animated short is, simply put, not to animate it at all. Rather than actually creating a new set of cel animation drawings, you can simply take some existing piece of clip art and just sort of... move it around on the screen. It doesn't have to look realistic; in fact, the more obviously fake, the funnier it will be.

While forms of this have been around for as long as film, it was until recently mostly associated Terry Gilliam's sequences in Monty Python's Flying Circus and the subsequent films. Now, however, with the explosion of web-based Video Collage, the techniques have become democratized, and entire new genres based on it have arisen.

Anime & Manga

 * A Ninja High School collection CD released in 1995 did this to the first story arc of the series.
 * Some of the witch sequences in Puella Magi Madoka Magica look like this.

Film -- Live-Action

 * Birdemic has a lot of this in its bird effects. Sadly, it was meant to be serious.

Live Action TV

 * The introductory sequences in Monty Python's Flying Circus, and many of the shorts in it as well.
 * The opening titles of Desperate Housewives.
 * Good Eats does it on occasion as a Shout-Out to Terry Gilliam, Monty Python being one of the biggest influences on Alton Brown when he was conceptualizing the series.
 * The Ron James Show has the Li'l Ron segments animated through clip art.

Video Games

 * One of the levels in Killer7 has cutscenes done in this style.
 * The entire game of The World Ends With You is done in this style... but done so well it barely counts. Plus, the room they saved was put to good use with one of the best soundtracks on a DS game ever.

Web Animation

 * Ikusa No Yukai, a Magical Girl parody produced by the fictitious animation studio Fujimira Doga.
 * JibJab is famous for doing this for political parody.
 * Kagaya Hime Miho-Chan: another Magical Girl satire by Fujimura Doga, played slightly straighter than other examples.
 * Animutations, Flash animations featuring cutouts of random characters and things usually lifted from Google Image Search.
 * Almost all Flash Poop variants of YouTube Poop are done that way.
 * The Homestar Runner animation "The Reddest Radish."
 * Also, the saleswoman in the Teen Girl Squad animation about the girls going to camp.
 * The Spongmonkeys, a pair of lemurs wearing bowler hats and sporting rather creepy-looking human faces that were once featured in a Quizno's sub shop commercial.

Western Animation

 * The Marvel Superheroes, one of the few uses not meant solely for comedy. It was like watching a comic book on TV.
 * Angela Anaconda - It's très interesting how they did this. They had models come in and take about 30 or so pictures for every mouth movement and a mouth movement for every letter in the alphabet. They then took the model's face and mouth movements and created each character.
 * Parodied in the "Badly Animated Man" shorts on Raw Toonage: the titular character is "animated" in this manner, while every other character is done in Disney's typical fluid style.
 * South Park was originally done like this, at least for the pilot - today, it's done in CGI drawn to resemble this style.
 * MAD, when spoofing real people.
 * All of the land animals from Fish Hooks.
 * One episode of Arthur featured a parody of South Park where Arthur Read is kidnapped by aliens and Buster Baxter is crushed by their flying saucer.
 * The storybook characters in Super Why!.
 * Wonder Pets
 * Mike Jittlov used this technique in many of his early works, most notably the first half or so of "Fashionation". There's even an example in the feature-length version of The Wizard of Speed and Time, with a tour bus passing Hollywood's famous Chinese Theater.