I've heard the question too many times when introducing someone to anime: "Well, if this is Japanese animation, then why do all the characters look white?" What most people are asking with this question is really something closer to, "Why don't they look Japanese?" It's time to answer the question once and for all.
Anime characters can look like elfs, mosters, robots, freaks, angels and whatever other fantastic creature the imagination can conjure up. Another question I've heard asked (in the same vein) is, "What's the deal with the big eyes?" Obviously the assumption here is that since the animators are Japanese, they would most likely animate eyes that look Japanese. This is a severe limitation to place on such a diverse medium, and usually it is a question that only Westerners are concerned with.
Firstly, Japan has a wonderfully unique culture that is a combination of the ability of the Japanese to incorporate the sciences and art of outside influences into a Japanese society that is literally an island. That island becomes an incubator for ideas, and often outsiders are confused because they judge all things Japanese without considering the context of Japan.
In fact, the 15th century already saw the Japanese using just drawings to convey a story. While most of the world can't think of animation without simultaneously thinking cartoons and children, anime is not a genre to the Japanese but rather a medium for communicating a story. It is small wonder, combined with the freedom that animation allows, that anime can take on such a myriad of shapes and flavors. When plot, setting and character are completely without limitation, race need not necessarily be as large an issue as it has historically been in reality.
It all started in post-WWII Japan; Hiroshima and Nagasake are just more examples of how there is nothing in the world that quite resembles Japan. They have already been through the apocalypse, and it is evident in their society and art. Shortly after the war, in an action that is representative of the way Japan incorporates outside influences, Osamu Tezuka (the father of Manga) was searching for a new medium. He found it because of the popularity of Walt Disney cartoons at the time. It was the popularity of the medium that was the strongest influence. Japan was no stranger to the drawn story.
The most obvious link then between a Disney and anime character is the big expressive eyes, but that doesn't necessarily mean that anime eyes are a holdover from Disney. Possibly even more so than other cultures, for the Japanese the eyes are the windows to the soul. When you watch your next anime piece, check out the difference between 'good' and 'evil' characters. Good characters have the large eyes that can convey sensitivty and emotion while the badguys are often portrayed as being soulless. Other times, the badguys will have large eyes and very small pupils, the idea being that not enough light has reached their souls. Anime is just trying to do the same thing as every other artistic medium: move you. We as humans respond readily to the language and expression conveyed through the eyes. It is an automatic reaction.
The imagination is limitless, so is the amount of money to make dreams a reality. Faced with financial restrictions, the Japanese utilized "tricks" such as moving two still shots across one another to convey movement, saving the expense of creating hundreds if not thousands of more prints. The animators instantly communicate dozens of complex emotions through the eyes alone. To continue some conjecture, it is also quite possible that the severe individualization of characters in anime (such as the wild hair or the unique eyes) could be a product of a country where everyone has the same color hair and had to wear uniforms through school.
If I may be so bold as to claim a point in all of this, it would be that the amazing variety that anime contains doesn't necessarily denote a caucasian race, if it in fact conveys race at all.
I include this solely to be complete and thorough, because in reality I think it detracts from the piece above. Upon returning from Korea my friend told me how some of his Korean friends commented about the size of Japanese eyes. Apparently they are the largest eyes of the South-East Asian races, or at least that's a belief that was held. The remark was made with a voice of envy. Could pride in such a difference be a factor? I'll leave it for you to think about.