Before reading the information below I would recommend:
The Celestial Sphere And The Equatorial And Horizon Coordinate Systems 

The mount refers to both the legs and the mechanism that holds and points the telescope. Having decided on a particular telescope design based on whether you are interested in observational astronomy or astrophotography or both, a mount has to be chosen accordingly and is also dependent on the type and weight of the optical tube which it has to support. Whereas any style mount can be used for observation, not all are suitable for astrophotography. It is important to know that the load capacity limit which is stated in the mount specifications, usually refers to visual use. It would be wise to divide this stated load capacity by 2 for astrophotography purposes. The load of your equipment will be calculated to include the optical tube, eyepiece, diagonal, finderscope,camera, barlows, extenders, auto guiding equipment…essentially everything that you will put on your mount for astrophotography. It should be noted that the balancing weights are not included in this calculation.

Digital photography requires perfect mount tracking and therefore the mount needs to have a dual axis motor (Right Ascension and Declination). Also, the mount should be equipped with a handset by which the mount can be made to automatically point and track celestial objects, some of which are too faint to be seen with naked eye. Very importantly, the mount should have an auto-guider port and periodic error correction capabilities. If the mount is of the equatorial type, a built in polar scope or an optional accessory such as Polemaster, would be needed to perform polar alignment.

An overview of the features to consider would be as follows:

Portability
Although the Equatorial Mount is heavier than an equivalent Fork Mount of the same specified payload, the fact is that it can conveniently be disassembled into more manageable parts which lends itself to greater portability.

 Versatility
The Fork Mount is versatile with respect to being easy and fast to setup for either observing or imaging. This mount is used most often with the shorter tube Catadioptic type telescopes such as Schmidt Casegrain and the Celestron Edge as this mount is not as stable generally as the GEM.

The German Equatorial Mount is also versatile with respect to observing and imaging but is more difficult and takes longer to setup. This mount would be a better choice however if you own different telescope types and weights, due to its increased stability over the Fork Mount. 

Movement:
An Equatorial Mount can be used to follow an object by moving the telescope for the most part, in only one direction, the right ascension. The Altazimuth Mount requires movement in an east-to-west direction (azimuth) and slightly up-or-down (altitude). If automatic tracking will not be used, the Equatorial Mount is preferable as it requires less adjustments which interfere with the viewing experience.

Eyepiece position:
When aiming and viewing objects overhead, the equatorial mount and refractor telescope combination can be very awkward to use and may involve crouching down very low to the ground to view an overhead object. The use of a diagonal with the refractor improves the eyepiece position. Using a Fork Mount for an overhead position would have the eyepiece positioned between the forks, not allowing access to use the eyepiece at all. Also regarding convenience, if the telescope is very large, the use of a stool or ladder to gain access to the eyepiece is necessary while smaller telescopes require kneeling or bending.

A Catadiotric, short tube telescope type, has the eyepiece higher off the ground but the diagonal allows a more comfortable eyepiece position.

The Newtonian telescope on an equatorial mount must be rotated in its tube holding rings to allow comfortable viewing position of the eyepiece when the position of the mount is changed. This applies to objects overhead and in other positions as well. This is in contrast to the Dobsonian setup which is always in a convenient viewing position.

Astrophotography:
Long exposure astrophotography greater than 20 or 30 seconds requires automatic tracking to prevent star trails and image rotation. Equatorial Mounts use to reign supreme for this task. The Alt-azimuth Mounts now can be bought with dual motor drives to provide the tracking requirements for astrophotography. In addition some cameras today can compensate for field rotation and send tracking messages to the mount.

Motor Drives:
Precise tracking is a requirement for astrophotography. In addition a correcting system (Periodic Error Correction) in the mount is needed whereby the error due to gear movement is corrected automatically. Also the ability to be slewed during the tracking process without having to reset the system is important.

Visual Tracking:
A motor drive especially for visual observation at higher magnifications is a welcome. Manually tracking  lends itself to possibly more frequent adjustments than with automatic tracking. Each time the mount is moved it takes time for the view to settle down detracting from the pleasure of the view. Therefore a motor on the Ra axis is welcome. If viewing for a while and depending where the object is located in the sky the Dec axis will have to be adjusted but less frequently. This can be done manually. The best ease of use would be a dual motor drive mount which would be completely automatic.

Stability:
If the option of a permanent observatory is not possible then the next best would be to have the most stable, moveable mount possible. The Equatorial Mount is the best choice in this case, especially when required to hold larger telescopes because even though it is the heaviest of mounts, the component parts can be disassembled lending itself to comparatively better portability. The main reason for choosing this mount would be the degree of stability when imaging. New types of dual drive, digitally controlled Dobsonians could also be considered for imaging, as they are very convenient and highly stable.

Finding objects:
A Go To Mount is definitely an asset. Otherwise, as much as 90 percent of the time is taken to find an object. The Go To feature can usually be added later or it can be included with the mount.

Construction and setup:
For ease of setup and portability, especially for larger sized telescopes, a Dobsonian cannot be beaten. Astrophotography is possible if the mount has a dual drive with error correction. with this type of mount. The Equatorial mount is not the only choice for astrophotography today. Fork equatorial dual motor drive mounts also make imaging possible. Some mounts of the Fork or Equatorial type can also be operated in the easy altazimuth mode for viewing.

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