Anime is the term used for the
Japanese style of
animation that is based on the
Manga (
comic book) style. Many popular Anime shows come from
Manga.
The most noticeable difference between Western animation and Anime is the unique Japanese manga and anime style, which is distinctive and fairly easy to recognize in all its variations. This is not to say the style is limiting. Within this broad common stylistic ground, each manga artist's technique is distinct. The stereotype is of characters with huge hair (in unreal colours) and large eyes, but there are many variations. Several theories have been put forth about the reason that Anime characters tend to have polychromatic hair, including one theory that states that it is a symptom of the Japanese belief that hair colours other than black (the most common hair colour in Japan) are far more appealing. However, I personally subscribe to the theory that it is because Manga are drawn in black and white, and so a Manga artist would get extremely bored colouring in peoples' hair on every page, and so they left some hair blank because they knew people would realise that the hair was still black. Eventually when the Manga were converted to Anime (in colour) the animators simply used colours other than black for some hair, possibly because people weren't used to seeing the characters with dark hair.
Perhaps the most important thing that sets Anime apart from Western animation is its broad appeal. Anime appeals to everyone, not just children. Japanese manga-ka (manga writers) write for everyone from small children to old men (there is even a category for ex-juvenile delinquent mothers). But even the children's stories tends not to be as simple-minded as the Western versions. Children's manga and anime shows in Japan will sometimes depict death; while the Western stories (on children's TV) seem determined to run away from such realism.
Another large difference between Western animation and Anime is the relative lack of superhero stories in Anime. Anime may put its protagonists into strange situations, but it frequently casts the main character as a (slightly) normal person reacting to events. Because of the general portrayal of characters as more normal than Western comic characters (they actually have lives outside of fighting evil -- see The Tick), much of Japanese manga and anime includes scenes of students in class or doing homework, or of people working in their offices. The work ethic seems omnipresent in the background, and in this way Anime reflects the traditional Japanese national character. Although the characters and their day-to-day lives are outwardly normal and mundane, they often interact with a more fantastic world in which anything can happen. It is this blend of fantasy and reality that can make Anime so appealing to people used to Western comics.
Anime characters tend to be more three-dimensional than their Western counterparts. They have flaws and annoying habits, and they don't always do 'the right thing'; sometimes they are just as selfish and small minded as the rest of humanity. Anime characters often have goals in their lives that give rise to the themes around which the Manga revolves. Anime and Manga are 'character oriented' whereas Western comics are more story oriented; in the Western style characters are often forced into actions against their personality because of the demands of the story, in Anime the story comes from actions that grow out of the character's personality. At least part of the reason for this is the way in which Manga are written. Manga are normally produced by at most two people (as opposed to a committee or a different writer for each storyline as it can be in Western comics). This means that the creator has a lot of control over the story, and so can write stories that fit the characters.
It is worth noting that Anime tends to show things in a far more complex way than Western comics. For example, in Anime the villains normally aren't just pure evil; they have their own hopes and dreams, they have reasons for their actions; in short they are just as well characterised as the heroes. Also, Anime tends to show the consequences of a mistake made by the protagonist instead of coming up with some trite solution to the problem; for example, if the protagonist doesn't help to defend his friends then the villain could kill one of them, and they could be permanently dead - no miraculous resurrections (at least not all the time).
Perhaps more important is that Anime characters change with time; they grow up and learn new things and react differently. They are not as static as most Western heroes. The villains can improve and redeem themselves. Life does have meaning and purpose, though it must be fought for. Hard work will pay off, but maybe only in the long run. Difficulties occur, but they can be overcome. Strength is found from helping others, even to the point of self-sacrifice. Not all stories have these spiritual or philosophical messages, but many do.
Finally, Manga and Anime can end. Heroes and heroines die, or get married, or disappear. They don't drag a story on for so long that it loses its sense of purpose. Many Manga are planned with a story arc over which the plot occurs and, at the end, the story finishes. It is rarely left open for a sequel. Manga tend to have one of three endings: the hero wins, the hero dies (usually after winning), and the hero sort of wins (but at a great loss).
Of course, many manga/anime do not share these characteristics. Some are as shallow as Western animation at its worst, and some Western animation embodies all these traits. This is just a discussion of Manga and Anime in general terms.
Update
I previously left unmentioned that physics are rather more negociable (read as: they are wierd) in anime. This leads to phenomenons such as bulletproof nudity and hammerspace (and, indeed, pretty much everything in Dragonball Z), whereby the laws of physics are suspended for visual impact (read (at times) as 'eye candy).
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This is part of my NodeSchool plan - node all those essays I write in school and have hanging around my hard disk.