The most common telescope mounts are the German equatorial mount, the fork mount, and the Dobsonian mount. Almost all telescopes sold have one of these types of mount, but there are a few other designs which are less common but still used with a variety of instruments.
Above: A typical alt-az mount, TeleVue's Gibraltar
While the Dobsonian mount is an altitude-azimuth (alt-az) mount, it works by friction alone, moving on Teflon bearings and using precise balance to hold the telescope in place for observing. Other alt-az mounts use a similar arrangement but are used for refractors or other optical designs and are therefore not technically Dobsonians. Others move freely then lock into place one the telescope is positioned. TeleVue's Gibraltar and Panoramic mounts are common examples of alt-az mounts. Even a regular old photographic tripod is a type of alt-az mount. Fork-mounted computerized Schmidt-Cassegrain telescopes and similar designs are also alt-az mounts but are not included here since they fall under the category of Fork Mounts.
Alt-az mounts are easy to use, especially for terrestrial observing. Depending on the type of mount, they may be difficult to use for astronomical observing. For example, photographic tripods are notoriously difficult to use for stargazing, while something like the TeleVue mounts or Vixen's Porta mount are very simple to use. The main drawback to an alt-az mount is that it will not track automatically.
Above: A typical split-ring equatorial mount, JMI's NGT-12.5
Modeled after the mount used on the 200" telescope at Mount Palomar, the split ring equatorial is a bit like a cross between a Dobsonian and an equatorial mount. Most often this is seen on large-aperture Newtonian telescopes. A ring runs under and along the sides of the optical tube, but the complete circle is broken at the top (the split-ring) to allow the optical tube to swing up and down. The ring itself is perpendicular to the polar axis (parallel to the celestial equator). Motion along the ring allows for east-west movement and motors turn the ring for tracking. Motion up and down through the split in the ring allows for north-south movement.
While this design allows for a Dobsonian-like telescope with automatic tracking, it has the disadvantage (compared to a true Dob) of the eyepiece position rotating as the telescope is moved around the sky. This necessitates a rotating upper tube assembly (where the eyepiece and secondary mirror are located) which makes keeping the optics collimated (properly aligned) more difficult.
The parallelogram is a type of mount normally used for binoculars. This design allows the binoculars to be aimed somewhere in the sky, then moved for an observer of differing height while keeping the object in the field of view. The long arm keeps the observer away from the tripod, making viewing more comfortable. It is also counterbalanced to make the motions of the mount very simple. In essence it is a type of alt-az mount. For large models especially, there is no better way to mount binoculars.