There are essentially two types of diamond cuts: brilliant and step.
Brilliant cut diamonds are rough diamonds that were cut and polished in a form that gives them exceptional brilliance; they have mostly triangular or kite-shaped facets that radiate from their center whereas step-cut diamonds have square or rectangular shaped facets which are long, narrow, and parallel. When cut well, a diamond will be luminous; cut poorly, it will be dull.

Brilliant Cut Diamond

Step Cut Diamond
A crash course on the modern manufactured diamond cut
Today, the preferred diamond cut is set by the proportions and standards of the Gemological Institute of America. But the GIA didn’t pave the way. A Belgian-born gentleman named Marcel Tolkowsky, who was just 21 years old and boasted of being a mathematician, physicist and an engineer, set out to tell the world how to cut a diamond. He did all that in a 1919 paper called Diamond Design – A Study of the Reflection and Refraction of Light in a Diamond. (For the real rabbit hole enthusiast, you can view the published paper here: https://tinyurl.com/3r8t26s4 ) He effectively created the theory behind the round brilliant diamond, with the aim of being the most beautiful in terms of brilliance, fire and dispersion. He ultimately suggested a set of proportions which should be followed when cutting a diamond to make it the most beautiful. His suggested proportions would be further improved on by the GIA, resulting in the round brilliant cut diamond, considered the highest standard in terms of brilliance, fire and dispersion.
When we refer to the way the diamond is cut and polished to the form we see when we visit a jeweler, we are talking about diamond anatomy. Specifically, diamond anatomy refers to the elements of Proportion, Symmetry, Polish, Fluorescence and Facets.