"Anime" is one of the very few examples of gairaigo, Japanese words of foreign origin, to be reassimilated by its original language and found in common usage.
Anime in Japanese
Like many modern gairaigo describing new technologies, the word originated as the transliteration of an English word, in this case "animation", which became アニメーション in katakana or animeeshon in romaji. Due to the differences in English and Japanese pronunciations, this word was very cumbersome (having six mora where most native Japanese nouns top out at about four) and was shortened in common use. This gave birth to anime (アニメ). The words are equal in the same sense that "SCUBA" and "Self-Contained Breathing Apparatus" are: the more convenient and natural-sounding shortened word has all but replaced the clunky and artificial-sounding original in normal conversation.
At the time, "anime" had pretty much exactly the same meaning as the word "animation" in English. It still does today. When Japanese children ask an American visitor "What kind of anime do you have in your country?" they aren't necessarily referring to Sailor Moon or Yu-Gi-Oh! -- they're thinking The Flintstones or Disney. Expressions such as "amerika no anime" (American cartoons) or "nihon no anime" (Japanese cartoons) are both equally valid noun phrases in Japanese.
Anime in English
It is is not exactly known how or when "anime" surfaced in English, but it is assumed that it, along with manga (漫画), was brought by fans of Japanese animation when it first began to penetrate Western Culture.
As an English word, it has several meanings. According to the geeks, anime refers to animation made in Japan (as a noun) or to something having the style prevalent to animation made in Japan (as an adjective). Because Japanese animation contains so many disparate works, it is impossible to make a more specific definition.
According to groups such as the entertainment industry, people who make dictionaries, and basically the rest of the world, anime is perhaps best defined as it appears in Merriam-Webster:
a style of animation originating in Japan that is characterized by stark colorful graphics depicting vibrant characters in action-filled plots often with fantastic or futuristic themes (1988)
Source: Merriam-Webster Dictionary (Online Edition) accessible at http://www.m-w.com/
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Yes, the layman's description requires action-filled plots and vibrant characters. Yes, it is the very Pokemon, the Dragon Ball that anime connoisseurs try to escape from. Yes, it is children's anime, shounen anime, or at the very very best, something by Studio Ghibli*, Evangelion, or Utena. Sadly, in practice, the vast majority of anime made and shown in Japan also fits this description.
* With the obvious exceptions of Grave of the Fireflies, Ocean Waves, Only Yesterday, and probably Whispers of the Heart.