There are two main categories of mount - Equatorial and Altazimuth.
Equatorial (EQ) mounts are designed such that one axis of rotation is aligned with the Earth's axis of rotation. In this way, once a target is acquired, it can be tracked by adjusting only one axis, or a motor can be used to rotate the axis at sidereal rate (the rate of rotation of the Earth with respect to the stars) EQ mounts can also be used to help locate objects using celestial coordinates (Right Ascension and Declination)
The main type of EQ mount used in amateur astronomy is the German Equatorial Mount (GEM). This is a relatively light weight mount that uses two axes mounted in a "T". A counterweight is used to balance the weight of the telescope tube. At the introductory end, these GEM's come in different sizes, often called EQ1, EQ2, EQ3, and so on. The EQ1 mount is little more than a toy, and should not be used with any scope. The EQ2 is stable enough for very small, lightweight telescopes, the kind I wouldn't recommend as a first scope. The EQ3, though considerably heavier and pricier, will be much more sturdy with any scope.
Fork Mounts are used on a number of computer controlled scopes, such as the ETX series from Meade, and the legendary Questar. In order to use a fork as an equatorial mount, it must be mounted on an angled base on a tripod, known as a wedge. These are considerably heavier than a GEM, and usually come on larger Schmidt-Cassegrains and Maksutovs.
Split-Ring or Horseshoe mounts are found on many large observatory scopes, and some high-end amateur scopes, but not on any entry-level scopes.
An Equatorial platform is a means of adding a third axis of rotation to a Dobsonian mount, to allow it to track. Some are expensive to buy, but often less than a premium GEM mount. They can also be home-built with moderate carpentry skills.
Mounts that rotate vertically (in altitude) and horizontally (in azimuth) are somewhat more intuitive to use, but when tracking celestial objects at higher powers, they require adjustment in two axes, not just one. If the motions are smooth, this is not as difficult as it sounds, and can be done easily with little practice.
Tripod mounted Alt-Az mounts come in a variety of styles. Typical AZ1, AZ2 and AZ3 mounts can be fiddly to use, and are more suitable for terrestrial observing and spotting scopes. A small EQ mount, like the EQ2, can be oriented as an altazimuth mount, providing a greater range of motion than one sold as an inexpensive AZ mount. I don't recommend the EQ2 as an EQ mount, but as an AZ mount, it works fine. Go figure. There are more premium Alt-Az mounts, such as the Unistar from Universal Astronomics, and the cradle mounts from Orion and Televue. These, however, are more than a beginner usually wants to spend.
The Dobsonian mount is essentially a large, wooden fork mount that sits directly on the ground. It is cheap, sturdy, and is able to support long tube scopes with minimal vibration. They are very sensible if you are interested in that type of telescope. In the image above, you can see my 8" on a Dobsonian mount sitting on an equatorial platform.