There are two basic kinds of telescope mountings: altazimuth and equatorial. Both will follow the stars and compensate for the Earth's rotation, but an equatorial mount is much better for astrophotography. That doesn't mean it is impossible to use an altazimuth mount for astrophotography, but the maximum length of your exposures will be limited.
"Altazimuth" is a contraction of "Altitude" and "Azimuth". An altazimuth mount moves in two axes - altitude and azimuth. Altitude tells how high above the horizon an object is, and azimuth tells its direction around the horizon, much like the pointing of a compass. You can point a telescope at any object in the sky by moving an altazimuth mounting in these two directions. You can also follow an object as it moves across the sky due to the Earth's rotation by moving an altazimuth mount in altitude and azimuth at the same time. This is complicated, so usually a computer controls the altazimuth mount.
A Dobsonian telescope is also an example of an altazimuth mounting. Some Dobsonian telescopes have special devices that let them track the stars, but most don't. Most Dobsonian telescopes track objects by observing them visually in the eyepiece and manually pushing the scope along by hand. It is almost impossible to take long-exposure astrophotos with a non-tracking Dobsonian mount. You can, however, take pictures of the Moon with a Dobsonian.
An "Equatorial" mount also can point at any object in the sky by adjusting two axes - the polar axis, and the another axis at a right angle to the polar axis called the declination axis. Like an altazimuth mounting, any object in the sky can be found by moving the telescope with these two axes.
The big advantage to an equatorial mounting is that once the object is located, it can be followed as it moves across the sky by moving only the polar axis. This greatly simplifies tracking an object to compensate for the Earth's rotation. Only a simple motor and gears are needed to move the polar axis, you don't need a computer at all.
If the polar axis of the telescope mount is accurately aligned with the axis of the Earth, a process called "polar aligning", an object in the sky can be followed for long periods of time with just the simple motion of the polar axis of the telescope. This allows much longer exposures than with an altazimuth mount, and produces better pictures.
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