Basically, a cabochon is a gemstone that has been shaped and polished. It is a very old method of cutting gems. As in, it’s what was done before faceting was developed. Traditionally, the top portion of the cabochon was smooth and domed. Though some had carvings on the top (think cameos). In modern times, a cabochon may be faceted on the top portion of the stone but the bottom is still flat.
(I’m providing pictures so this makes more sense)
Technically, cabochons are not really "cut". They are shaped and then polished. It’s much easier to produce a cabochon than it is to cut a faceted gem with many faces.
The shape is often round or oval, but I’ve seen and even used other shapes.
If a gem has very good color, but isn’t transparent enough to be faceted, it can still be shaped and polished into beautiful cabochons. It’s also common to cut softer stones as cabochons since gems with a hardness score of less than 7 (on the Mohs scale) can easily be scratched by the process of faceting.
This technique is also often used for opaque stones. Certain stones are almost always cut "en cabochon", including opal, turquoise, onyx, moonstone, and star sapphire. Jewelers often use the term "cab" when referring to a cabochon.
Another fun fact: The term comes from the French caboche, meaning knob or small dome.