So, Mailyn has already explained how the anime were categorized: into types rather than genres. In fact, this way, it's easier to be more specific about what kind of anime it is. However, manga are usually classified by the age and gender of the target audience and this is basically the following are the "genres."
Oh yeah, I think I should give an explanation of how the manga industry works in Japan first. So basically, almost all manga are serialized in magazines that can be weekly, bi-weekly, monthly, bi-monthly released... and these magazines will contain one chapter per issue, maybe two if there is a break taking place. All these magazines are sold very cheap and so almost everyone buys it. Then, as the manga is serialized in magazines, tankoubon are also released. Tankoubon are the book edition of the previous chapters and so each tankoubon contain between 5-10 chapters. Since the magazines are so cheap, they are usually recycle and readers by their favorite manga in tankoubon, which are also very cheap. There are usually no difference in the storylines between the magazines and tankoubon; however, some modifications can be made to the drawings. (esp. if the mangaka was in a hurry for the magazines, s/he will take time to re-draw for the tankoubon).
So here we go:
++ Shonen are manga usually featuring male protagonists and are characterized by high action. This genre of manga is aimed at an audience composed of young/teenagers/young adult males, although it can still appeal to many females. Some of the most popular shonen manga in the past involves sports (soccer, football, basket ball) or fighting (samurai, pirates) as well as mecha.
Magazines: Weekly Shonen Jump, Shonen Sunday...
Some titles: Bleach, Naruto, Hikaru no Go, One Piece, D-Gray Man, Hunter x Hunter, Eyeshield21, Prince of Tennis...
++ Shojo are basically girls' manga with female protagonists and a lot of romance, still aimed at teenager/young adult female audience. It also covers a wide range of subjects from historicals to fantasy. But be aware that some shojo manga are more graphically explicit than others, i.e. they contain sex scenes.
Magazines: Hana to Yume, Ribon, LaLa, Margaret, Sho-comi...
Some titles: Mars, Ouran High School Host Club, Basara, Fruits Basket, Hanakimi, Cards Captor Sakura, X...
++ Seinen are manga aimed at an older and more mature male audience. These manga are usually less action-based and have a wider variation of art styles and subject matter.
Magazines: Ultra Jump, Afternoon, Big Comic...
Some titles: 20th Century Boys, Monster, Akira, Ghost in the Shell...
++ Josei are manga drawn mostly by women aimed at a more mature women audience. The stories tend to be more realistic, based on everyday living experiences of women in Japan and contain more realistic romances instead of idealized ones. The art style is also more restrained and realistic.
Magazines: Chorus, CUTIE, Young Rose...
Some titles: Gokusen, Honey and Clover, Kimi wa Petto...
++ Kodomo are manga for kids and the plots are usually more family- orientated and feature more child-friendly themes and concepts.
Magazine: CoroCoro Comics, Comic Bonbon
Some titles: Doraemon, Hamtaro, Pokemon...
++ Yaoi are manga that features gay/homosexual relationships. This genre is highly popular with the female audience. Some of these manga are more graphically explicit while others are subtle relationships. The yaoi genre is further separates into shota where one of the protagonist is very young, and shonen-ai where the relationships are subtle and not consumated.
Magazines: BxB Gold...
Some titles: Haru wo Daite Ita, Cut, Silver Diamond, Hybrid Child, Bronze...
++ Yuri is the female version of yaoi, i.e. that the manga features lesbian relationships. Sorreh, but I'm not completely familiar with this genre ^^;
++ Hentai is basically drawn pornography... need more explanations? :P Again, not familiar with this genre...
++ Doujinshi are fan-art or self-published manga. Most of the time tho, doujinshi mangaka will take characters of pre-existing manga and draw stories for fan-service. Altho some mangaka draws their own doujinshi :P Anyway, these doujinshi are very popular with the fans and can be very expensive, since they are drawn in limited quantities. Obviously, as you can guess, there are no magazines for these.
So this basically covers all the genres. Altho manga usually have target audiences, the audiences are not restricted to the genre that correspond to their age or gender. What I mean is that these genres are not exclusive, you can love anything... I'm a girl and I love shonen, shojo and yaoi manga as you'll see. All that matter is really that you find the stories/plots good, interesting and entertaining and that you appreciated the art :) That's one of the extra tho, the art :) Some mangaka have incredible talents and their drawings are soooo beautiful! Thanks God there are art books, which are collections of the best art and color pages by a mangaka or of a manga.
Just a note of warning, the two main genres of manga are shonen and shojo. Very often, seinen manga will be categorized as shonen and josei, as shojo. So don't be surprised if you never see seinen or josei... ppl, esp. foreigners, rarely allude to seinen and josei :P
So I hope I didn't bore anyone and I hope I was clear and that it will help the newbies :D