Telescope mounts can be even more important than the telescope itself. A wobbly mount that cannot handle the weight of the telescope mounted on it will make observing with your new telescope a frustrating experience and will waste the abilities of your telescope.
These are a special class of alt-az mounts. Usually they are made of wood and they tend to me the cheapest mount available for beginner telescopes. This means you tend to get more telescope for your money with a dobsonian mount. They swivel on their base and let the telescope rock up and down, and are robust and easy to use. They can be hard to use when viewing things near the zenith (straight up) due to their design, and they tend to be heavier and more cumbersome than other types of mount.
Dobsonian telescope mounts are the easiest for beginners to get used to as their movement is easy to understand and they are stable and robust. This is our recommended mount for most beginner telescopes.
This is the type of mount you find on many astronomical telescopes. They rotate at a special angle so that they can follow objects as they move across the sky as the earth moves. They need more setting up than other mounts and can be more difficult to get used to. They tend to be more expensive than dobsonian mounts and cannot handle such heavy telescopes.
These are necessary for much astrophotography as they are needed to allow the telescope to follow the correct path across the sky by moving only one axis. To do the same with a dobsonian or alt-az scope would mean moving it in two directions.
Unless you have a particular need for an equatorial telescope mount (such as astrophotography) we wouldn't recommend these for beginners. They are harder to learn to use and the extra money spent on the mount would be better spent on the telescope at the budget of most beginners.
These mounts tilt up and down and rotate on a horizontal axis. Scopes on this type of mounting (that aren't dobsonian) tend not to be intended for astronomy and are often meant for wildlife viewing etc. They are similar to camera tripods and as a result look very similar to equatorial mounts. They are lighter than dobsonian mounts but cannot handle such large telescopes. This means they have the trade-offs of an equatorial mount with none of the advantages.
We wouldn't normally recommend this type of mount for an astronomical telescope, even a telescope for beginners.
Motorised mounts can be fitted to any type of telescope mount. Some mounts come with them and some can have them fitted later. Some cannot at all. They range from simple motors that let equatorial mounts follow a point in the sky (for easier viewing or photography) to fully computerised mounts that will point the telescope at particular objects in the sky for you.
These mounts tend to be much more expensive than their manual equivalents. Some can be fiddly to set up and they add a layer of complexity. On the other hand they can make finding objects in the night sky to view much easier. Opinions on them vary wildly from person to person.
Unless you have a very large budget we wouldn't recommend these for beginners. To fit within a typical beginners telescope budget you end up with a poorer telescope than with either of the manual mounts, and the compromise simply isn't worth it.