One of the simplest types is the altazimuth mount.
This allows you to move the telescope in altitude (height) and in azimuth (horizontally) and hence point at any desired object. With most cheap telescopes, you will find the telescope mounted onto a small fork. In addition, the telescope has on one side a small metal rod which is connected to the fork mount (this applies mostly to the smallest telescopes). This lets you fix the elevation of the telescope. The other telescope axis can usually be rotated 360° horizontally and can be fixed in position with a small screw.
Thus equipped, you can use this type of mount to point at and observe any object in the sky. This design has the advantage that it is very light to carry and requires no great technological knowledge to operate it. When observing astronomical objects, you must constantly correct the telescope in the two axes to compensate for the rotation of the Earth.
You will not have to make the same size of correction on both the axes: The Earth's rotation means that a star will ‘move’ in the sky by about 0.25° per minute. Objects appear in the east, rise in a circular path, reach their highest point in the south at the meridian and finally sink again until they set in the west.
So you will need to keep ‘recapturing’ objects with the altazimuth mounts again and again as they drift out of the field of view of the eyepiece after quite a short time. This is not easy when the telescope has a very simply built mount in which no fine position adjustment is possible. Such a mount in any case absolutely unsuitable for astrophotography, as a photographic object must always remain precisely stationary in the field of view. In addition, altazimuth mounts are affected by field rotation.
However, there are also easy to operate altazimuth mounts. These have two knobs called ‘slow motions’, one for each axis, and enable fine control of the movement of the telescope. This allows you follow astronomical objects much more precisely. If you are planning to buy a telescope with an altazimuth mount, I would advise you to get one with the ‘slow motions’.