The type of mount you choose to put your telescope on is a significant thing to consider. Other then the telescope itself, it contributes more to the overall experience then just about anything. Carefully consider what type of mounting system fits you and your situation best.
There are basically 2 types of mounts: alt-azimuth, and equatorial.
Alt-azimuth mounts are relatively strait-forward. They allow a telescope to move up or down, and rotate to the left or right, relative to the ground. This makes them fairly intuitive - simply point the scope in whatever direction you wish; the mount essentially allows free movement in any direction. Beginners, as well as seasoned astronomers, can appreciate the simplicity of this design.
A common variant of this type of mount is called a Dobsonian. Often referred to simply as "dobs", they are essentially the simplest type of telescope mount - basically a large, free-spinning stand that lies directly on the ground, eliminating the need for a tripod. They are fairly inexpensive, and can even be built from scratch relatively easily. They are often used with large reflector telescopes, which would otherwise be difficult / unwieldy to use.
The main and only real weakness of an alt-azimuth mount, is that it does not naturally track the night sky. As time passes, objects in the sky move in a circular pattern corresponding with the earth's rotation. This results in objects that you've found moving out of view. The more you're zoomed-in, the faster they appear to move away - and this in turn means more frequent adjustments. Whether this is a problem depends on you and your scenario. Casual observing, deep sky, or wide-angle objects do not pose much of a problem. Objects that require high magnification (such as planets) or astro-photography (where objects need to remain still) are both examples of when this can become an issue.
Equatorial mounts are designed for keeping objects in constant view. While slightly more complex, they provide an easier way to track objects as they move across the sky. With equatorial mounts, one axis is always pointed towards the north/south poles, along the same line that the earth (and sky) move. When this is the case, a telescope can be spun around one axis and follow the sky in perfect harmony. Only small, single adjustments need to be made around one axis; while alt-azimuth mounts need to be constantly adjusted both up/down and left/right.
A common variety are german equatorial mounts. These have been around for some time. They have a bar mounted perpendicular to the main axis. One side of the bar holds the telescope, while the other holds counter-weights, to keep the axis balanced. This allows the telescope to be rotated around the main axis smoothly and stably - without weight affecting its orientation.
Technology has advanced over the last few decades, and telescope mounts are no exception. Most telescope manufacturers now provide computerized tracking capabilities built directly into their mounts. This can make finding objects in the vast night sky much easier and significantly faster. These systems very a great deal; some simply passively "guide" your movements, while others use motors to move to objects you've selected. These systems require accurate date, time, and location information. Some types require this information to be manually entered, while more advanced models have built in GPS. These systems all feature substantial libraries of objects throughout the sky to find and observe.
These systems can also be built into most types of mounts explained above. However, more advanced tracking systems tend to be built into mounts with tripod designs, such as german equatorial mounts, as well as "fork" mounted telescopes. The fork style is relatively recent, and features a telescope (often a small cassegrain) mounted between 2 arms attached to a revolving base. Interestingly, fork-style mounts can be either alt-azimuth or equatorial, depending on whether the base is oriented horizontally (level), or towards the earth's poles.