Slate is the main kind, but the "non-slate" category is much wider. There are materials that are like slate, such as slatron and permaslate, and some have a plastic honeycomb surface or even particle board. Slate, and all of its alternatives, are covered in a felt play surface so you won't actually see it--but you will notice some differences over time.
Slate is a very durable, hard material which is perfect for pool tables because it can be leveled to a precision tolerance (within .0001 inches). In a game where the position of the pool balls is everything, it's important to make sure the balls won't be rolling randomly. Slate is also more resistant to damage over time, especially when the slate is thicker.
The main reason that the alternatives exist is the cost, as slate tables are comparatively more expensive. There is also a weight component, as slate is a kind of rock. Slate tables, therefore, are extremely heavy--especially three-piece slate tables, which are used in professional settings (bars have single-piece slate tables). Alternatives, the non-slate tables, are generally cheaper, lighter or both.
Unlike slate, all of the alternative materials cannot be leveled to precision. This means they may look flat, but aren't necessarily precisely flat. There is a possibility that a ball will roll the wrong direction, or it may not be completely level. The other reason players prefer slate to alternatives is the durability: These alternative materials will chip and wear over time.
Regardless of the surface you choose, make sure that your players never scratch the surface of the pool table with the cues, and when racking the balls, always lift the rack about 1/2 inch off the felt. Vacuum dust and other things off the table periodically, and cover the table with a decent, well-fitting cover when not in use. These precautions will help keep the pool table in good condition regardless of the type you choose.