Let's talk about rocks, baby ...
Quick vocab lesson first. The diagram below shows the parts of a faceted gemstone. I don't know that you'll be able to work this info into regular conversation, but knowledge is power.
What may be of more use to you, however, is the terminology around the way stones are shaped and how they're cut. Stones are cut and ground into specific shapes to enhance the way light reflects and refracts within them and to bring the focus to a specific section of the raw material. Cutting and finishing gemstones is known as lapidary or lapidary arts. The technology used varies from computerized faceting machines to sandpaper. There are large shop versions and table top or hand held wet saws and grinders.
Cabochons, or cabs, are flat backed stones. They're often the first type of stone jewelry students learn to set. Lately, a specific type of cabochon, druzy, has become very popular. They're essentially a collection of crystals on the surface of another stone. Here's more information on how they're formed. Mickey Lynn's website also has great information.
See also: this Etsy blog post about gem trends for this year. (Who gets paid to write about that and how can I steal their job?!?)
Faceted stones are often cut into standardized shapes and styles. Here's a few shapes:
The shape and number of sides are designed to refract and reflect light in ways that highlight the gem's color and beauty
The Princess cut and Round Brilliant cuts are two that many people are familiar with because it is commonly used for diamonds.
The Asscher cut was the first signature cut to be patented. Brillant, rose, princess, and radiant are all types of cuts that can be used with other shapes like round and square stones.
Rose cut have facets only on the surface of a cabochon.