Anime: Animation originating from Japan. Comes in a wide variety of genres, with certain kinds for children, early & late teens, and adults (often called Hentai.).
Shonen: Anime geared towards boys who are 10 to 18 years of age. Shonen often have a lot of fighting, friendships between men (not to be confused with Yaoi), and female characters with exaggerated features. Shonen series have a less "cutesy" look that Shojo series. Examples include Bleach, Fairy Tail, and Death Note.
Shojo: The opposite of Shojo. Series in this genre are made for girls 10 years to 18 years old, much like a Shonen series. This is where the similarities end, however. Shojo series focus on things like emotions, as well as human and/or romantic relationships. Shojo also differentiates itself from it's male counterpart by looking more "cutesy". Examples include Ouran High School Host Club, Vampire Knight, and Romeo X Juliet.
Seinen: A subset of Shonen. Seinin is meant for men who are at least 18 years of age, all the way to 30 years of age, although some series can be geared towards men who are well into their 40s. The big difference between Seinen and Shonen is that Seinen series have more realism in them, and have a larger amount of subject material to use. Examples include Chobits, Ghost in the Shell, and Akira.
Josei: A subset of Shojo, and counterpart to the Seinen genre. The first difference is the targeted age group. Josei series target girls from 15 years of age, to women at least 44 years old. Josei stories are mostly about the everyday living experiences of women in Japan, although there are a few that detail experiences in a high school setting. Josei also distances itself somewhat from Shojo series by toning down the cutesy look, yet retaining some features, such as exaggerated eyes. Examples include Honey and Clover, Loveless, and Paradise Kiss.
Harem: A loose genre in which a single protagonist is surrounded by at least 3 members of the opposite gender. A common example of a Harem is a single guy surrounded by (beautiful, sexy, or cute) women. A woman surrounded by a bunch of guys is called a Reverse Harem. In recent years, the requirement of the opposite sex has been taken out, allowing for Yaoi or Yuri harems to be created. This genre has gotten some flak in recent years due to the fan service (see below) being pushed out. Examples include Ai Yori Aoshi, Love Hina, and Omamori Himari.
Magical Girl: Often belonging to the sub-genre of fantasy anime, Magical Girl often focuses around a single girl, or a small group of girls, who have been given magical powers from some higher being. Magical Girls have an item that allows them to transform into their magical form, and also have a magical mascot of some sort. Magical Girls often defend Earth from the forces of evil, or some other dimension. Examples include Sailor Moon, Hyper Police, and Tokyo Mew Mew.
Mecha: A genre that covers all series that feature the use of controllable robots. Said robots are often massive in size, although there are a few series in which the robots are of a much more humbling height. Examples include Gigantor, Voltron, and Full Metal Panic.
Yaoi: Also known as Boys' Love, or BL for short, Yaoi is a female oriented genre in which men engage in homoerotic relations with one another, some of which are of a sexual nature. In such a relationship, there is the seme (attacker or pitcher) who follows the uke (receiver or catcher). Yaoi can also be mixed with other genres as well. Examples include Gravitation, Kaze to Ki no Uta, and June.
Yuri: Counterpart to Yaoi, Yuri is also known as Girls' Love, or GL for short. Yuri focuses on female/female relationships, and like Yaoi, are sometimes sexual. Yuri can focus on things like sex, spirituality, or emotions. Examples include Revolutionary Girl Utena, Aoi Hana, and Read or Dream.
Hentai: A genre that is geared purely towards adults. The Hentai genre is devoted explicitly to sex. Some Hentai deals with straight relationships, while others have gay relationships. Some Hentai is consensual, while others have rape or degradation as a central element. Examples include La Blue Girl, Angel Blade, and Dragon Pink.
Fan Service: Not an actual genre, Fan Service is often an element in many genres. Fan Service involves upskirt panty shots, naked butt shots, boob shots (naked or otherwise), boob grabbing, or anything else naughty & dirty (not necessarily dealing with sex). Examples of shows with Fan Service include Kekko Kamen, Cutey Honey, and Rosario + Vampire.