Doujinshi can mean several things. Most generally, it is an unofficial comic book (or other art, possibly digital), usually by an amature, that is based on characters not "owned" by the writer/artist. Somewhat like fanfiction. Depending on who publishes these, they can range from legitimate to completely illegal, usually the latter.
The second definition follows; the thing is, most Doujinshi are that way because they are about subjects that wouldn't be covered in the actual series--that is to say, adult topics. Again, these range from somewhat more mature versions to outright pornography, and from legit to illegal (almost always the latter). Note that despite being illegal, these are generally tolerated in Japan.
The third version is collections of (mostly erotic) amature art distributed in "underground" channels via disk (floppy, that is, and now the Internet/usenet). Once in a while one of these artists will gain some noteriety and their works may make it to a CD ROM collection or something of the sort.

This definition is in context to Anime. See more contextual defintions for Doujinshi.