Otaku are people who are obsessed with a hobby to the extent that it becomes a major part of their identity. Otaku (おたく) has a somewhat negative connotation in Japanese. Much like the English word "nerd", "otaku" implies socially awkwardness.
A person can be otaku about practically any hobby or interest. The most common types of otaku in Japan are:
1. Anime / Manga Otaku
Virtually everyone in Japan has read manga or watched anime at some point. However, some people are obsessed with it.
2. Cosplay Otaku
Cosplay Otaku are into wearing costumes and role playing. In many cases cosplay otaku are highly social and gain friends through cosplay activities. They may spend large amounts of money attending events and buying costumes. In many cases, they learn to make their own costumes (sewing etc..).
3. Game Otaku
Game Otaku spend much of their time playing games. They usually establish social connections in virtual worlds with people they may never physically meet.
4. Idol & JPop Otaku (Wota)
A Wota is obsessed with female or male idols (often JPop idols). For example, the JPop group AKB48 has a musical theatre in Akihabara in which they perform nightly. Wota collect idol magazines and posters and often seek to meet or photograph idols in person.
5. Figure otaku (Figure moe zoku ~ フィギュア萌え族)
Figure Moe Zoku can be translated as "figurine lover gang". It's a term for collectors of Anime / Manga figures. These figures are often highly realistic.
6. Train Otaku (Tetsudou Otaku)
There are plenty of trains in Japan. Tetsudou Otaku are obsessed with photographing trains and/or riding trains. They may also be interested in train uniforms and model trains.
7. Robot Otaku
Japan invests far more than any other country in robot research. Many Japanese robots are frighteningly advanced.
Robot Otaku are interested in popular culture related to robots. They may also take interest in robot research. In extreme cases, they're involved in building robots (as a hobby). There's a shop in Akihabara that sells nothing but robot parts.
8. Pasocon Otaku
Pasocon Otaku are obsessed with computer technology. They study hardware and software specifications in detail. They enjoy discussing the latest devices. They may spend large amounts of time configuring and customizing software. In some cases, they become experts at computer programming or network security.
9. Wapanese
Wapanese comes from the English "Want to be Japanese". They are non-Japanese people who have a obsessive interest in multiple aspects of Japanese culture. They may develop an incredibly positive view of Japan that's somewhat unrealistic. Some become skilled at Japanese language and/or martial arts.
10. Female History Otaku (Reki-jo)
Reki-jo are female history buffs. They're interested in pre-industrial Japanese history. They view this period as an ideal age of innocence and adventure. Reki-jo often form social groups that gather to dress in period costumes. They may adapt the speech and mannerisms of old Japan.
11. Voice Actress Otaku (Seiyuu Otaku)
One of the many sub-types of Anime Otaku. Seiyuu Otaku are obsessed with the voice actors and actresses from Anime productions.
12. Military Otaku (Gunji Ota)
Gunji Ota are obsessed with military uniforms, hardware and war stories (often in manga form).