A doujinshi (同人誌) is an especific type of manga (japanese comic) drawn by amateur authors and/or fans. Despite the existence of original stories in this genre, most of them revolve around manga and/or anime that already exists. So this medium, just as fanfiction, allows the author to explore aspects of the work they're paying tribute to, that the work itself doesn't.

A good number of these doujinshi are pairing-based. This means that the manga/anime being not centered around the romantic aspect of its characters (or addressing it just superficially), the doujinshi-ka likes to give his/her favourite couple more time together, re-tell some scenes or tell a different story altogether with their romantic relationship as the main theme. Or just mix two characters together for the sake of smut crack shipping.

Aside from the whole series or pairing diversity, doujinshi also come in different sizes, ratings and prices. The usual size for a standard doujin is B5, being A5 the next most commonly used. Ratings vary greatly, from G to hard core hentai or yaoi. So be careful about that. As for prices... it all depends. If you live in Japan, you'll probably find doujinshi cheaper than anyone else. If you don't... well, let's just say that from $5 and up, it's all game.

So, to sum it up, doujinshi is a beautiful and rather rare form of art. Us, crazy collectors, are doomed to day and night digging, last-minute bidding and spending absurdly big amounts of money in them. I'd recommend to avoid it and run fast. Otherwise, the doujin-bug will caught you and never let go.