A shoot-em-up ("shmup" for short), is a video game where the player has limited control of their character and the focus is almost entirely on annihilation of their enemies. There are several major sub-genres.
Fixed shooter
A fixed shooter has players only able to control their two-dimensional position on the screen and sometimes the direction they are facing. Fixed shooters have various degrees of movement freedom, sometimes only allowing control on a one dimensional line. Also known as a proto-shmup, examples include classics such as Space Invaders and Galaga.
Scrolling shooter
The scrolling shooter usually has the players auto-moving throughout a level at a mostly fixed speed. These are the games most purists would consider "shmups," other games listed here are related sub-genres of the larger shooter classification These generally fall into two categories, "vertical" and "horizontal". Vertical shmups come into two more varieties, so-called tate or yoko shmups. Tate shmups are those that are played on a monitor oriented vertically, and such games generally exist solely in arcades. Yoko shmups are those played on a standard horizontal monitor set-up. Horizontal shmups are always played on horizontally oriented screen, although certain exceptions, such as Darius, use more than one monitor to create an extremely long playfield. Examples of horizontal shmups are Gradius and R-Type; some famous vertical shmups include Raiden and Ikaruga.
A third class comprises of games where players move through a "tube", essentially a 2D scrolling shooter plane extended to three dimensions. Movement is allowed throughout the vertical plane perpendicular to the player's forward movement. They may be pseudo 3D sprites as in Zaxxon or fully 3D polygon environments as in Star Fox or Duke Nukem: Manhattan Project
Rail Shooter
The rail shooter is typified by a character travelling along a preset path while movement input is prohibited, or restricted to small variations such as changing speed, choosing forks in the path or small momentary movements off the path. Gameplay usually consists of aiming at targets as they appear along the level, so rail shooters are often played with light guns, especially in arcades. Examples include House of the Dead and Panzer Dragoon.
Run-and-Gun Shooter
This is a hybrid of the platform game and the scrolling shooter, consisting of a character running along the ground armed with a projectile weapon. These games generally do not include auto-scrolling; the screen will move as the character progresses. Most games are played horizontal, from a side-scrolling perspective, but several notable exceptions are top down, some even with screens oriented vertically. Some famous examples are Metal Slug and Contra (a game that consists of both side-scrolling and top down levels).