So here's a long one. (Pardon me for its length and its wandering; I was doing chain-of-thought writing :)
Many people might say "Manga are Japanese comics, and Anime is the Japanese version of animation. Anime is usually, but not always, the animated version of popular manga." That's partially true, but it can be misleading. (Note that "anime" in Japan technically means any animated film, and "manga" is any printed cartoon, but people in the rest of the world take them to mean animated films or comics from Japan.)
First of all, though an outsider might think Japan "stole" comics from the West, this is not true. Japan has been making cartoonish art for a very long time (there are humorous ink drawings of animals and caricatured people from hundreds of years ago, bearing striking resemblances to modern manga). True, some aspects of manga are taken from the West (Osamu Tezuka, the "father" of modern manga, was influenced by Disney and Max Fleisher), but its main features, such as simple lines and stylized features, are distinctly Japanese. It may be that Chinese art had more influence than Western.
(Also, speaking of China, I should note that Anime is now a general Asian phenomenon, not just Japanese. I understand there are many fine works of manga and anime being produced in many places around the world. However, as far as I understand, the roots are in Japan, and Japan is still considered, at least here in the US, the center of the anime world. This may well change in the future.)
Secondly, Japanese manga and anime come in all types, for all sorts of people. Unlike the U.S., which generally seems to believe that "comics are for kids" (though this has been changing recently), Japanese manga-ka (manga writers) write for everyone from innocent young children to perverted sex-starved men (there is even a category for ex-juvenile delinquent mothers!). But even the kiddie stuff tends not to be as simple-minded as the American versions (not including intelligent American comics, but more thinking of TV shows). Children's manga and TV anime shows in Japan will sometimes depict death --- while the U.S. (on children's TV) seems determined to run away from such realities of life (note how the U.S. version of "GoLion" ("Voltron") deleted all references to one of the protagonist's death). And, not surprisingly, 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. Manga and anime also tend to protray technology sympathetically, while some U.S. comics seem almost to avoid it, or revile it, or simplify it as much as possible.
A third major difference is the unique Japanese manga and anime style, which is distinctive and fairly easy to recognize. This is not to say the style is limiting. Within this broad common stylistic ground, each manga artist's technique is distinct and unique. The stereotype is of characters with huge hair and large eyes, but there are many, many variations, from L. Matsumoto's seemingly unevenly drawn squash-shaped "ugly" protagonists, to the soft-edged figures in Miyazaki's work. And, of course, there is less emphasis on the "superhero" world of the U.S.. In most manga, the men and women aren't necessarily exaggerated extremes of their gender stereotypes, and they wear things other than skin-tight costumes. In fact, manga and anime characters tend to have unique and aesthetic tastes in fashion. (It's also true that many modern U.S. comics have thankfully broken this stereotype, and serious-matter cartoonists like Alan Moore or Art Spiegelman have always been around.)
And one minor difference between Japanese manga and general superhero comics like D.C. Comics or Marvel Comics (aside from the black and white nature of manga), is that manga are usually the vision of a single writer (though editors have a large say, and sometimes direct the story). Unlike the general superhero type, where many writers tend to do different plots and stories, manga are more like novels, complete and detailed worlds that are the vision of a single author. The characters remain consistent, and they are allowed to grow and develop. On a related topic, manga also tend to be drawn for a weekly or biweekly publication containing numerous other comics by other authors --- and the editors expect cliffhangers/you-really-want-to-read-the-next-issue endings each time. So the plot HAS to develop and HAS to be interesting at a fairly rapid clip. (There are, after all, crowds of hopeful would-be manga-ka waiting in the wings).
(One last difference is the onomatopoetic characteristic of the Japanese language; sound effects fit in much better, and look less stupid, than in English comics. This is just a facet of the language; translated manga sound effects also don't work as well.)
Perhaps it is the mix of harsh reality with the tantalizing world of fantasy that makes Japanese manga and anime so appealing. Many popular series, such as Doraemon, Ranma 1/2 and Kimagure Orange Road, follow the lives of seemingly ordinary people --- they go to school, do homework, get reprimanded by parents --- who have a shadow life that makes them somehow special, whether by psionic talent or friends who are rather different (robots from the future, or aliens from other worlds). I suppose all this serves to allow the reader to sympathize with the characters, and yet escape from bland, normal daily life to a fantasy world that is far different.
Even in worlds that exist in the far future, or long ago, the reader is drawn into a 3-dimensional character, one who is far from perfect, one who has stupid little habits or major character flaws --- and who has hopes and dreams that the reader can sympathize with. Unlike some American super heroes, who often seem to just go around defeating Evil (as wonderfully spoofed in American comic "The Tick"), Japanese characters usually have other goals in life that play large themes within their lives. I heard recently the characterization that manga and anime are "character oriented." The more I think about, the more I think this is the right description. Characters aren't forced into plots, like a foot into a too-tight shoe; instead, stories grow out of the characters. The heart of manga and anime is in the hearts of the characters.
That brings us to three other aspects of manga and anime that I really like: the reality of the world, the spirituality, and the fact that things end.
With comics, the merging of art and words creates a unique medium. The art pulls in the mind, and the words make the reality. A picture may be worth a thousand words, while words may convey what art cannot, but the two types together are truly powerful. As for Anime, animation can do inexpensively what special effects crews couldn't even touch until the recent rise of computer graphics. Art is a limited form of virtual reality. Art, however, requires plot to make a story come to life.
As I've said, even children's Japanese comics and animation deal with things like death. They also show that one's enemies aren't Just Evil. In series like Gundam, you can see that the enemies have hopes and dreams of their own, and do, in fact, have reasons for what they do. They aren't just crazy, or just plain evil. They're real.
Actions have consequences. If the protagonist screws up, he or she has to deal with those results ... and, if the person is smart, he or she will remember not to make that mistake again! The characters grow and change, learn new skills, get better at old skills, mature and gain wisdom (unless, of course, it's a comical series like Doraemon :)
Another trait of manga and anime that I have always liked (though perhaps I hadn't realized it until recently) is their tendency to contain a sense of spiritual optimism ... and not just simplistic good-over-evil stuff. Bad people can improve and find redemption. Unhappy heroes can find themselves, through personal crisis, and in doing so find happiness. 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. And when these simple but universal themes are woven more or less convincingly into the fabric of good plots and characters, magic happens.
And finally, like all good stories and all real stories, manga and anime have a tendency to end. Heroes and heroines die, or get married, or disappear. The anime series are especially good about this. They tend to have one of three endings: the hero wins (the throne, the person of the opposite sex, whatever), the hero dies (usually after winning), the hero sort of wins (but at a great loss). Of course, the anime or manga is often carefully crafted to either jerk tears out of your eyes, or make you stare in wide-eyed absorption to the very very last line of the credits. I can't describe it here, but think of the ending to any truly good movie, and you probably have it.
I guess I've wandered quite a bit over this topic. I also probably displayed a bias for semi-serious manga/anime (which I prefer), and I also probably didn't quite describe the nature of certain genres (such as pure business manga, or sex manga, or the purely political humor comics). And, of course, I'm sort of glossing over the fact that there is LOTS of trash out there. Like any field, manga and anime have their lemons, the ones with no plot, 2-D characters, truly tasteless jokes, and artwork from hell. However, the best manga and anime are true gems that should not be missed --- little portals into other worlds that will entertain, educate, and delight.
---Text Copyright 1995 Eri Izawa; further revisions in 2001, 2002, 2004, 2005 rei [at] mit [dot] edu)
(For more information on manga and anime, please see my Anime and Manga Page.)