The most common type of shooting games are horizontal and vertical. Vertical STGs are games where the screen scrolls forward or backward(like in Touhou). Horizontal STGs are game where screen scroll to the left or right. There are games (but it's rare) that use both styles. Good example is 3D game Ether Vapor where screen changes from vertical to horizontal and even isometric view.
Shooting games usually feature solely the player character fighting against the entirety of the enemy forces. Most shooting games are split into several stages, with each stage ending with a fight against a boss before one can progress to the next stage. In addition, while the enemies fought may take a lot of punishment before they go down, the player character generally requires only one hit to be destroyed, adding to the challenge.
Examples of STGs include the Touhou series, Galaga, Gradius, DoDonPachi, ESP Ra.De., Mushihimesama, Twinbee, Psyvariar, Shikigami no Shiro, Ikaruga, Radiant Silvergun, Salamander, Einhänder, R-Type, Giga Wing, Blazing Star, Zero Wing, and many others.