Derek Yu (the Aquaria artist) has been a very busy bunny lately. He's recently published an explanation of what Doujin Shoot 'em ups are and his personal top ten.
As a quick recap...
Shmup is short for "shoot 'em up", which applies mainly to spaceships blasting away other flying objects on a technical 2-D playing field (I say technical because some games like Ikaruga use 3-D graphics). Doujin is a Japanese term used for small publishers, somewhat similar to the word "indie" in English-speaking sections of the world. So a doujin shmup is generally a 2-D flying shooter made by a just a few people or a small software house.
Soot 'em ups have always been something of a mystery to me so this small break down helps considerably. Unfortunately my total lack of twitch gaming skills holds me back but I will be trying at least a couple of the games mentioned in this article some time soon.