Animation is the use of drawings or other movable objects to create interesting visuals, stories, cute shit, and in some (japanese) cases, mind-breaking horror.
Webster's Dictionary defines "animation" as "an animating or being animated/the preparation of animated cartoons", so Webster can go get "dolphin raped". However Crimmas' Dictionary of Words that Mean Shit defines "animation" as "images of a stationary/inanimate object in sequence to present the illusion that the object is in motion". We all have a different mental image in mind when we hear that word. I think of Popeye. You may think of Bugs Bunny. The syphallitic homeless man that screams about bacon on your way to the subway probably thinks of Drawn Together.
But animation in this respect isn't limited to cartoons. Of course the word itself means "stuff that all, like, moves and shit" but specifically we're talking about the illusion of animation. Anything that you can move or alter can be used to create the illusion of animation. Wanna make it look like your bed got up, waddled to the gas station and bought some beer? you could animate that with a stop motion camera and a complete lack of anything worthwile to do.
Simple. Let's say you want to create a short video of the Enzyte guy waving (hand or dick, your choice), and you want to use stop-motion. Just make a figure that looks like the Enzyte guy, rig up a camera (screw it onto something) and take a picture. Then move his hand/meat bat a little, and take another picture. Now do this 70 more times. Process it at the traditional 24 frames per second, and congrats! At the standard frame rate, you just created a whopping 3 seconds of animation.
Who was filled with enough self-loathing to subject themselves to this the first time, you ask? Well that was J. Stuart Blackton in 1906, who created a short film called "Humorous Phases of Funny Faces" presumably inbetween twirling a handlebar mustache and clubbing the Irish.
After that was "Fantasmagorie" by Emile Cohl which I assume was about the strange side effects from taking this newfangled "heroin" stuff for a headache.
Then Windsor McKay made this mindfuck that nobody talks about but was, in actuality, the first of the cartoons as we know them today.
Of course it's hard to influence the world and change American culture forever with something that makes people scream and flee the building, so two years later in 1914 he made "Gertie The Dinosaur", a story about a lovable giant lizard that eats trees and kills elephants for shits and giggles. This was essentially the cornerstone for cartoons as we know them today.
After that was Krazy Kat, Felix the cat, and Mickey Mouse.
Then Betty Boop, Popeye, Snow White, Porky Pig, Bugs Bunny, Tom and Jerry, Gumby, The Flintstones, Hanna Barbera ruling animated television with an iron fist and a well-guarded budget for the next 30 years and then animation had a collective orgasm.
In the 80's there were plenty of cartoons that we still remember (and make into terrible movies) today, but most of them were also just toy tie-ins. The 90's were like woodstock for cartoons.
The Simpsons, Ren and Stimpy, Rugrats, Beavis and Butthead, Duckman, South Park, Daria, The Critic, King of the Hill, Dexter's Laboratory, Powerpuff Girls, Dr. Katz, Aeon Flux, Celebrity Deathmatch, holy motherfuck.
Today, Seth McFarlane is setting up a legacy similar to Hanna Barbera and Klasky Csupo (or however the fuck it's spelled), who created several Nicktoon series . Only problem is TV critics aren't necessarily on his side and Hanna Barbera never had the balls to make a new show entirely devoted to making The Flinstones black, so only time will tell.
Basically you've got Cell Animation, Stop-Motion, and Digital Animation/CGI/etc.
Traditional Cell animation is what probably comes to mind when you think "cartoons". The most common way it's produced is by painting the moving parts of the image onto acetate and laying that over the stationary parts in the background, then shooting the frame. However this type of animation became obsolete recently, and has been replaced by Digital Cell animation.
Anime is done using this method, and they're by far the best at it. You may have heard of Akira, which has been credited with kick-starting the boom of Anime among western audiences. It's been called the greatest achievement in animation, because A: it predates digital techniques by a couple of years, combined with B: Just goddamn look at it.
Stop-motion usually involves clay or movable foam figures. It's often referred to as "Claymation" although apparently that shit's copyrighted so everyone says stop-motion now. It isn't limited to clay or foam but it's usually some sort of 3-dimensional object being moved around. Celebrity Deathmatch used foam with clay heads, Robot Chicken uses action figures, that White Stripes "Fell In Love With a Girl" video used legos, which somehow didn't result in any suicides to my knowledge.
As far as computer-generated animation goes, obviously the first thing you're going to think of is Pixar. Toy Story was the first full-length movie made entirely in CG. Some of you may also remember Reboot.
Flash animation has been growing in popularity as well. Although some claim it looks rubbery and unauthentic, it has been used in interesting ways recently, and is a fuckton cheaper to use as well as quicker than most other methods.