Cutting gemstones is an extremely delicate art. It takes a lot of patience and skill. The process is much like sculpting but generally much smaller and more precise to ensure the gemstone has the proper shape and absolute best possible sparkle.
The rough minerals are first cut into the general size and shape the cutter is aiming for on the final product. From there the cutter begins to shape the rough stone into the desired shape by cutting the general shape on a lapidary then moving on to the facets. The last step is polishing for that shiny gemstone brilliance. Each shape and cutting style have very specific facets needed to create the best possible brilliance.
These are just the standard shapes. There are also cabochons which have a smooth rounded top, checkerboard surfaces, concave, bubble cuts, and many more new shapes and variations being created every day by artisans all over the world.