There are lots of different types of telescope mounts and many manufacturers have designed their own. While they look quite a bit different they pretty much comply to either one or the other of the two major types of mounts.
Type 1: The Alt-Azimuth mount telescope
This is the simplest type of telescope and the easiest to use. It moves your telescope in two directions - up/down and side to side. Pretty much as easy as that. And it is very intuitive because anyone can easily learn how to use a telescope with this kind of mount. This type of mount is often seen on inexpensive telescopes because it is the simplest to use and the simplest to make. To use a telescope mount like this you simply move the scope up/down or left/right to point at whatever object you want.
Type 2: The equatorial mount telescope
This type of mount is bit more complex and a bit harder to use. This picture here shows the mount under the white tube of the telescope. There are two different axes on this type of mount. The long silver bar is one axis and if you were to grab that bar with your hand and twist it the telescope would rotate. The other axis is parallel to the body of the telescope (there is a black square under the scope) This axis points directly at the north star. And the black box is a motor. It will turn the telescope one revolution every 24 hours. This negates the movement of the earth and you can point the telescope at a night sky object and the motor will slowly move the telescope so the object stays in view.
Equatorial mounts are far superior to alt-azimuth mounts in that they will make it much easier to stay on a particular object in the sky. And they are better for taking photos because with a long exposure it is easy to keep the object exactly in the same spot. But they are a bit harder to use and it takes some time to get the knack of the unusual way you have to move the scope to point it at things.
Here is a video of me with the telescopes mounted on the two different types of mounts. It shows you how they move and the benefits/deficiencies of each type: