All gemstones (which includes diamonds) come in a variety of shapes and sizes. Cutting gemstones is actually an art called lapidary and while there are different methods to the trade, I'll be focusing on faceting. Faceting is done with machinery and is meant bring out the brilliance of a gemstone because the bottom reflects light so that when you view it looking down you can see the light. Faceting creates small geometric flat planes and it's what is typical when thinking of an engagement ring or wedding ring. Faceted gemstones are commercially the most popular by far and I know personally I enjoy using faceted stones in our rings because of the brilliance it can show off of a stone.
Types of cuts
There are many types of cuts but most are a variety of two main cuts. The first is the brilliant cut. Brilliant cuts are triangular and its facets spread outward from the center of the stone. The second is the step cut. Step cuts are rectangular facets that appear as steps because they ascend the crown of the stone. I use both brilliant and step cut gemstones as well as rose cut gemstones. Rose cut gemstones are faceted gemstones but with a flat (cabbed) base.
Gemstones typically are seen in a few common shapes including round, oval, pear, square, and rectangular. Shapes don't actually portray the cut of a gemstone though because cuts are really the combination of both the shape and type. However, in the industry certain names have become associated with both a shape and cut. For example, a princess gemstone is a square faceted cut stone. You wouldn't be wrong to call it a square faceted cut though because that's what it is. An emerald cut is nothing more than a rectangular step cut.
What's a good cut then?
Good is subjective a bit here because "good" cuts aren't the same for all gemstones. The objective of a good cut is to display both the brilliance and color of a stone in its ideal state. Color saturation and light reflection are two ways to determine if the cut is good. If you look at a stone in different angles with a light shining on it, is there equal reflection? Light reflection shouldn't be confused with a stone flashing though. Ultimately it's a matter of preference because good cuts are subjective. If you're happy with the look of the stone then for you, you've got a "good" cut.