There are two major forms of animation: 2D (hand-drawn, or computer assisted) and 3D (CGI). This page focuses on 2D animation.
The overall process of creating a 2D animation is divided into three parts: pre-production, production, and post-production.
Pre-production involves establishing the plot of the completed animation. Because each shot in animation is so labor-intensive, it's imperative to get shots right during pre-production. Re-animating a shot (because a character uses knowledge not gained until later in the story, for example) is very expensive, especially compared to live-action filming.
Most animations are storyboarded, in which the main action in each scene is drawn in a comic book-like form. Storyboards are usually pasted on large foam-core or posterboard sheets, which can be quickly read through by the staff. During pre-production, the staff reviews the storyboard for consistency, and parts of the storyboard may be redrawn multiple times.
In North America, the dialog for the animation is recorded during pre-production. It's then provided to the animators so they can draw mouth movements to precisely match the dialog spoken by the actors. In other parts of the world (most notably in Asia), the dialog is recorded during production as the actors follow along with the rough animation.
Each shot in a 2D animation involves multiple single drawings of characters(although some computer programs such as 'Anime Studio', can create characters much like puppets, which can be posed at different keyframes, and the computer moves the character between those keyframes). To demonstrate the concept, think of a still camera that can take many photographs in rapid succession. Imagine aiming that camera at a person and taking many quick photographs as the person walks towards you. Each photograph corresponds to an individual drawing in an animation.
A shot may contain only one drawing (especially in "limited animation" forms like anime), but usually contain about ten to twenty drawings. These drawings are divided into "keys" and "in-betweens." Keys are important drawings that convey the extremes of the character's movement--the first drawing is almost always a key--while in-betweens (or "tweens") are the remaining, less important drawings in the shot.
Typically, each drawing begins with a sketch, in pencil, of the character's pose. This drawing is then cleaned up in ink, though this is not always done; Disney went through a period in which this step was skipped, as is most clearly visible in the original 101 Dalmations. Skipping the inking step was thought to make the animation look more vibrant and alive, but critics and audiences didn't like the look. 
This is the point at which computers often step in. Some companies hand-ink each drawing, writing over the cleaned pencil lines with a pen. Others--especially studios whose artists can draw very clean pencil lines--will scan the pencils directly into the computer, then ink the drawing digitally.
The drawing is now considered a "cel." Before computers, the drawing would now be photocopied onto a sheet of clear celluoid or acetate, then hand-painted. This created a "cel," which was placed on top of a background painting and filmed with the click of a film camera. Cels are still highly prized by collectors, though few true cels remain. Many of those on the market are reproductions.
Today, the digital cels are colored on a computer. Meanwhile, a background is drawn (on the computer, or hand-drawn and scanned). The background is imported into an animation program, and each cel is layered onto the background in succession. This is then saved on the computer as a single video file.
In practice, the keys are usually drawn by a top animator, and these are then scanned and assembled into a "rough cut" of the shot while the keys are given to a junior animator, who will draw the tweens. The film can then be assembled during production, and junior animators can be directed in small adjustments to improve the flow of each shot and scene. In Asia, this is typically when dialog is recorded, so that the junior animators can match the tweens to the vocal performance.
Even after all the tweens have been colored and added to the animation, the film is far from complete. There's still music and sound effects to be added, as well as visual effects (glows, hazes, etc.). The animation also may need to be edited in the more traditional film sense; shots cut slightly short or held longer, even scenes cut entirely. It's said that one famous anime series (Gundam Seed) changed its ending when the primary actor broke down during recording, and new animation was hastily drawn, colored, and incorporated into the final minutes of the show.
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