There are two very general categories for polished gemstones: faceted and cabochon. Cabochons come in many different shapes, but what really defines a cabochon is that it has a smooth, domed surface. A faceted gemstone, meanwhile, has a surface covered in tiny faces, or facets.
A cabochon is often the cut of choice for opaque and semi-transparent stones, in order to better show off their color. This style is excellent for displaying desirable inclusions, as in rutilated quartz. Below is a variety of cabochon gemstones in different shapes.
Cabochons also emphasize asterism (a star-like illusion on the dome of the cabochon) and chatoyency (a cat's eye or band of light phenomenon; shown below) in certain gemstones.