Anime catagories and styles

Shōjo (Girls Anime) & Kodomo (child)::

Bishōjo (beautiful girl): A term that features pretty girl characters. This is a "Major Catagory" that others sometimes fall under.
Examples: Magic Knight Rayearth or Negima.

Mahō shōjo (Magical Girl): Stories about Mystical Girls or females possesing magical powers.
Examples: Sailor Moon or Cardcaptor Sakura.

Reverse Harem: Features one female character that is romantically involved with many male characters.
Examples: Ouran High School Host Club or Fruits Basket.

Bishōnen (beautiful boy): A term that can be used to describe any anime that features "pretty" and elegant boys and men.
Examples: Fushigi Yūgi or most CLAMP shows.

Shōnen (Boys Anime) & Kodomo (child)::

Sentai (fighting team): It refers to any show that involves a superhero team.
Examples: Cyborg 009 or Voltron.

Robot/Mecha: Features super robots or machanical suits.
Examples: Mobile Suit Gundam or Mazinger Z.

Josei (young woman) & Seinen (young man):
I have combined these sections since they overlap on several levels::

Mahō shōnen: Stories about Guys possesing magical powers.
Examples: D.N.Angel or Fullmetal Alchemist

Harem: this genre focuses on a male character surrounded by the romance of multiple female characters. Typically, the male cohabits with at least one female. It is usually considered a Shōnen or Seinen series.
Examples: Ranma ½ or Love Hina.

Magical (Exotic) Girlfriend: These anime focus on the romantic relationship between a man and at least one woman of extraordinary origins such as alien, supernatural, or technological.

Post-Apocalyptic: Deals with a post-apocalyptic worlds.
Examples: Fist of the North Star or Nausicaä of the Valley of the Wind.

Moé: Anime featuring characters with perky, cute, weak, or naivè behaviors. In some way, they are not overly independent.

Expertise: Specializes with a specific topic in depth. Topics range from sports, the arts, and cooking.
Examples: Eyeshield 21 with football, or Yakitate! Japan with bread-making.

Ecchi (perverted): This is derived from the Japanese pronunciation of the letter "H" (Symbolizing "Hentai" but not the same), Sexual humor and fan service are prevalent.
Examples: Oruchuban Ebichu or He Is My Master.

Hentai (indecent sexuality): This term is synonymous to pornography or erotica, as hentai content specifically consists of such.
Examples: La Blue Girl or Bible Black.