That's SHoot-eM-UPs, scrolling shooters, whatever you wanna call them. The games where you've got a spaceship and you're shooting everything.
They've been around for quite a while, having origins in such games as Space Invaders, Asteroids and Yar's Revenge, proceeding through the 8-bit era with The Guardian Legend and Life Force, expanding and maturing through the SNES and Genesis era with games like Gradius 3, Thunder Force, Star Fox (in a way) and Axelay.
Then, with the next generation, things got moody, deeper plotlines and gameplay through games like Einhander and Radiant Silvergun, and continuing this trend through the last generation with R-Type Final and Ikaruga. Most recently, we have the rise of danmaku, or "bullet hell," shooters like Touhou and Mushi...Mushimasashiro... this game *shudder*, Geometry Wars, and the re-release of many of the classics of yesteryear through download services like Virtual Console.
So, anyone out there got any favorites? These kinds of games are a bit on the short side, so finding new ones is a constant thing. Personally, I've always liked Axelay, for its unique mechanics and weapons, and its trading between a behind-the-back view and a more standard side view, as well as its captivatingly different soundtrack; and I got R-Type Final over the weekend, so I'm messing with that. It's kicking my butt, if you must know. Just like a good shmup should.