Altazimuth
The altazimuth is the simplest type of mount. It has two types of motion: vertical (or altitude) and horizontal (or azimuth). These two motions allow you to point your telescope at any position along the horizon. Altazimuth mounts are generally good for low-power telescopes but should not be used for deep-sky photography, since it's hard to track objects with an altazimuth mount.
Dobsonian
Dobsonian mounts were first developed in the 1970s by John Dobsonian. They are a type of altazimuth mount. The Dobsonian mount is mounted on a heavy platform on the ground. Most often, the Dobsonian mounts are used with larger Newtonian reflector telescopes.
Equatorial Mounts
Equatorial mounts are used for watching the night sky over a long period of time and are thus very popular in astrophotography. With an altazimuth mount, if a telescope is trained on a particular star, as the earth rotates the star will move out of the telescope's view. However, an equatorial mount can much more easily keep track of the star, either manually or electronically.
German Equatorial Mount
German equatorial mounts are, as the name suggests, a type of equatorial mount. The German equatorial mount is distinctive with its large counterweight mounted on the opposite side of the telescope, on an axis known as the polar axis. The polar axis is aimed at Polaris, thus aligning the mount with the Earth's rotation. The mount is turned at the same rate as the Earth's rotation, allowing the telescope to keep track of the same object in the night sky. Both Newtonian reflector and refractor telescopes can be used with a German equatorial mount.
Fork Mount
Telescopes with shorter optical tubes tend to use the fork mount instead. The telescope is mounted between two arms that make up the fork and is connected to a motor. Unlike the German equatorial mount, the fork mount doesn't need a counterweight. The fork mount can be quickly converted from altazimuth mode (used for observing) to equatorial mode (for astrophotography), making it much more versatile than other types of mounts; however, it is also considerably heavier and less portable.