Hentai - This is MA-rated stuff. It shows intercourse, unlike ecchi. Example: Boku no Pico (Do not watch this, I WARN YOU.)
Historical - Anime based on a particular time period, very often the Edo/Tokugawa period in Japan where the shoguns ruled the country. Example: Nobunga Concerto
Horror - Despite the name, the horror anime isn't very scary, instead it implies alot of bloodshed and mutilation, I suppose a better word would have been gruesome. Example: Another
Josei - A very rare (at least in the west) genre aimed at the adult woman (ages 18-40). They are normally themed about living the life of an ordinary woman. Example: Honey and Clover
Kids - Like the name, it is anime suited for the younger audience. Example: Pokemon
Romance - This is a relatively popular genre which deals with the detailed creation and progression of the romantic relationship between two characters. Basically, two characters in love. Example: Clannad
Magic - A more 'kiddy' styled anime. (young girl with magic wand etc.) It can also be used to refer to animes with a 'realistic' sense of superpowers such as, results from praying, magic charms etc. Example: Sailor Moon
Martial Arts - Action Packed with the majority of the 'actiony' scenes from a form of martial arts such as boxing, kenpo or mixed martial arts. Example: Ranma 1/2
Mecha - Another action packed anime genre, but with the use of giant robots or Mechas. Often includes space, military and fantasy. Example: Date A Live
Military - Where the main character is often in the military or ex-military. Frequent use of military weapons and military combat knowledge. Example: Attack on Titan
Music - Where the theme of the show encompasses the art of music, either where it is used to express emotion or as a hobby. Example: Shigatsu wa kimi no uso
