Biography
Otaku come in many flavors, but one thing can be said for each and every one of them. They've each staked out their own favorite thing, and they obsess over it relentlessly. Regardless of other intelligence, an otaku will have an obsessive, unhealthy, and almost encyclopedic knowledge of their chosen topic.
There are almost as many flavors of this type of character as there are things under the sun, but a few of the major ones are:
Anime or Manga Otaku (which is what most people think of when they hear the term "otaku")
Cosplay Otaku
Gaming Otaku
Idol Otaku (wota)
Military Otaku
Technology Otaku
Gun Nut
Blade Enthusiast
Essentially, someone could be an otaku about just about anything: politics, sports, history, etc. When otaku is used by itself by a Westerner, 99% of the time it will mean "anime/manga otaku".
Neither geek nor nerd is an adequate translation. However, in modern use, both words may carry a shadow of the right connotations of obsessive interest and/or social ineptitude; see the geek page for details. Think of the older, more pejorative senses of geek and you're on the right track. The British term anorak and the Internet terms neckbeard or weeb/weaboo are also close translations. Speaking of The Internet, in more dickish online communities, autism-related terms get slung around in a similar manner. The closest troper-speak cognate would be "Loony Fan." In Japan, the term doesn't carry a positive meaning at all. One of the first things most Japanese language classes often have to teach people is that calling yourself an otaku in Japan is a very bad thing. (Although it must be said that in more recent years this sense is mellowing out, to the extent that more Japanese are self-identifying as otaku. See The Other Wiki's page on this for more info.)
A related term is hikikomori, which refers to a teenager or young adult who withdraws completely from society for an extended period, typically isolating themselves within their parents' house and becoming psychologically fixated on particular hobbies; hikikomori in media are usually otaku of some sort. Hikikomori are also critically viewed as lazy and outright creepy, which doesn't help the perception of otaku much — especially after 1989, when serial killer Tsutomu Miyazaki was shown to be both an otaku and hikikomori, leading to a moral panic.
Otakuism is associated with men, with the notable exceptions of the Fangirls, Wrench Wench, the Cosplay Otaku Girl, and creators of a certain kind of comic. However, female otaku seem to be either getting more common lately or becoming more relaxed about showing it.
The term itself comes from the very polite form of "you", which can come off as socially awkward. The best guess as to how the term became associated with obsessive fandom is that the word was an inside joke among the production staff of the anime series Super Dimension Fortress Macross in 1982, and that they would have characters (notably Lynn Minmay) use the over-polite form of address, even when inappropriate. Fans picked it up and used it in conversation with each other even well past the point when they would use other forms of "you", such as "kimi", "Anata", or "omae". A writer for a Japanese magazine noted the meme and wrote an article that cemented the term as being used for obsessive fans.
See also: Occidental Otaku, Chuunibyou. Compare: Loony Fan (who are more weird than obsessive) and The Movie Buff (a similar type of obsession over movies.) Prone to creating a Nerd Hoard of items relating to their obsessions.