Pile & Flat-weave
Rugs are the sort of thing that can honestly been divided into as many “types” as there are rugs. But that would be a useless classification system. Instead one can use very broad strokes and divide the whole of carpets into two types: pile carpets and flat-woven carpets.

A short-piled carpet close up.
Pile carpets get their name from the fibers that stick out of one side of them. These perpendicular fibers are called the pile. The pile on these rugs makes them much thicker, softer, and more insulative. This last characteristic is reflected in the fact that pile carpets more commonly originate from colder climates.
Flat-woven carpets are pretty much everything that isn’t a pile carpet. These are the rugs that lack the pile layer. Their fibers are densely woven together and, for the most part, run parallel to the overall plane of the carpet. When these flat-woven rugs are of Middle Eastern origin they are sometimes called kilims.

A flat-woven carpet close up.
Sometimes flat-woven rugs are falsely labeled as a lesser art, but these weavings are not to be dismissed. The patterns are often just as wonderful. In modern times pile carpets are sometimes made exclusively for export, while flat-woven carpets are made for the weavers personal use. This sort of situation can lead to the flat-woven carpets having superior patterns that better express emotion.