A sturdy mount is obviously important for visual observing--the lack of a quality mount is a primary complaint about cheap telescopes--but for astrophotography, the mount is every bit as important as the telescope itself. The mount's ability to hold the telescope steady during a long exposure (and in breezy conditions), as well as the tracking accuracy are important factors.
Tracking Accuracy and Guiding
Astrophotography of deep-sky objects requires capturing very faint photons, and that requires taking long exposures. Typical exposure times might be 5 minutes or more. However, if you just put a camera on a telescope with a tracking mount and take a 5-minute exposure, you will be disappointed to find that the stars all appear trailed. This is because most mounts cannot track precisely for longer than 30 to 60 seconds due to imperfections in the drive gears.
There are two ways around this problem. One is to "guide" the telescope. This involves a second camera, often looking through a second small telescope piggybacked on the main telescope, which takes exposures of a guide star every few seconds and sends commands to the mount to keep it tracking perfectly on that star. Meanwhile, the camera on the main telescope can now take as long an exposure as desired. This is the required method for telescope and camera setups that require exposures longer than about a minute.
Alternatively, you can keep the exposure time under the limit imposed by the unguided mount, typically around 30 seconds. The problem with this is that on most telescopes, 30 seconds simply is not a long enough exposure. The necessary exposure time depends on the camera to some extent, but mostly upon the focal ratio of the telescope. Just like using a camera lens for regular photography, a faster focal ratio (or f-stop) results in a shorter exposure. A telescope that is "slow," meaning a larger focal ratio than about f/5, will need to be guided. A "fast" focal ratio telescope, f/4 and below, may not need to be guided to get at least some pretty good pictures. A very fast system like the HyperStar lens for Celestron SCTs that gives a focal ratio of f/2, allow for excellent images to be taken without guiding.
Above: An image of the Dumbbell Nebula taken with a Celestron 11" telescope on an alt-az fork mount, using a HyperStar lens to image at f/2. This is a stack of ten 20-second exposures, so just over 3 minutes total time.
Alt-Az vs Equatorial Mounts
There are two main styles of telescope mount: altitude-azimuth and equatorial. Alt-az mounts are ideal for visual observing, and may be possible to use for some astrophotography. Equatorial mounts are better suited to photography and are the mount of choice for deep-sky imaging.
An alt-az mount moves a telescope in directions that are parallel and perpendicular to the horizon, so basically up-down and left-right. This is a perfect system for visual observing as the eyepiece stays in a convenient location as the scope is moved around the sky. However, in this configuration the object being tracked appears to rotate over time because of an effect called "field rotation." Because everything is the sky appears to move along an arc as the Earth turns, the orientation of an object rotates relative to the horizon. Think of the constellation Orion: when rising in the east, Orion is lying on his side; when highest in the sky 6 hours later, he is standing straight up. This rotation is not a problem visually, but for astrophotography it limits how long an exposure you can take, even if the telescope were guided. This limit is usually around 30-60 seconds.
Above: An alt-azimuth mounted telescope.
An equatorial mount has one axis that is aligned with Earth's rotation axis. In the northern hemisphere, this axis points toward Polaris, the north star. As Earth rotates, this axis of the mount does too, at the same speed as the Earth, once in 24 hours. This allows the telescope to track the sky without field rotation. Now by guiding, you can take as long an exposure as desired. There are two main types of equatorial mounts: the equatorial fork mount and the German equatorial mount.
Above: A telescope on an equatorial fork mount.
In many cases, it is possible to convert an alt-az fork mount into an equatorial fork mount by adding an accessory called a wedge. The wedge goes between the tripod and fork mount and tilts the fork arms of the mount at an angle based on your latitude, allowing them to point toward the celestial pole and track in just one axis. This eliminates field rotation. It also allows you to change between alt-az mode for viewing and equatorial mode for imaging. The drawbacks are that larger fork-mounted scopes can be heavy and difficult to mount onto the wedge, and the long arms of the fork make this type of mount inherently less stable than a German equatorial mount.
Above: A telescope on an German equatorial mount.
A German equatorial mount (GEM) provides the best stability and is thus probably the ideal choice for deep-sky astrophotography. A GEM has a counterweight shaft to allow the optical tube to be balanced about the polar axis. This keeps the lever arm from the axis to the telescope very short and stable. The balanced system means the tracking motors have to do very little work. Another advantage of a GEM is that you can easily switch optical tubes. If you have an SCT or RC for long-focal-length imaging and a small refractor for wide-field imaging, for example, you can easily use either scope on one mount.