Manga: On a basic level, Manga is nothing more that printed comics & printed cartoons. A big difference between Anime & Manga is that Manga is read by all age groups. Manga covers just about every subject there is in the world. Most Manga that is printed is in black & white, although there are a few series that are printed in full color. Many Animes have a Manga of some kind: some follow the show to the letter, some deviate a little bit, and others don't completely follow the show's story at all.
Dojinshi: Like Manga, but with a few differences. Manga is printed by big publishers, while Dojinshi is printed by amateurs (Although a few big time writers will do Dojinshi outside of their regular work). Dojinshi is based on original material, and those who base their work on big time writers do so in small quantities so as not to deal with litigation. Dojinshi that is based on existing material is sometimes a highly collectible item, as only the lucky or the fast will get to them first.
Shonen: *
Shojo: *
Seinin: *
Josei: *
Harem: *
Magical Girl: *
Mecha: *
Yaoi: *
Yuri: *
Hentai: *
Fan Service: *
*These terms are on the "Anime Terms" page. They can be (and are) applied to Anime as well, especially if a Manga is based on a show, or the other way around.