Diamonds are cut into many shapes--even more than the ten shown here-- and the shape impacts the value every time. It can also help date the jewelry to an extent as well, especially in conjunction with the setting.
The most popular and prevalent shape is the Round Brilliant. It's what comes to mind when someone says the word, "diamond," and it is used not only as the centerpiece of the majority of engagement rings but as accent stones as well. It is called round for the obvious reason, and brilliant because the facets are primarily triangular or kite-shaped-- check out the diagram if you're wondering what I mean. The great thing about a brilliant cut diamond (most of the other shapes are also brilliant cut as well) and the reason it is called brilliant, is that faceting like that causes an immense amount of sparkle.
In contrast, the Emerald cut next in line is a step cut. This means the facets are all rectangles and trapezoids arranged in steps, creating a window effect. The facets aren't set up to sparkle, but to reflect. It's an elegant look and very different from the round. An emerald cut is octangular, a rectangle with the corners cut off forming an octagon. This shape is, not surprisingly, used often on emeralds as well as other gemstones.
The Cushion is exactly that-- shaped like a pillow. Sometimes it's square like this graphic, other times it's elongated, but its corners are always rounded. The cushion is an older cut, a direct evolution from the Old Mine Cut, and had a resurgence in the past decade with the popularity of antique styles.
Next in the graphic is the Oval, named for obvious reasons and a very popular shape right now. Another brilliant cut, finding an oval that is not too long, too thin, too fat, or too short, can be difficult. When you find one that is just right, it's gorgeous and bright.
Heart shapes are uncommon and hard to find in appealing dimensions as well, often too wide or with an unnoticeable cleft. Again, when they are cut well, they can be quite beautiful, even if you're not a fan of hearts in general.
Radiant cut diamonds are brilliant cut square diamonds that have the corners mitered, leaving them as octagonal and very bright. Jewelers like them because they do not have pointed corners to break. Radiants are another shape that can be more square like the graphic or more elongated to appear like a sparkly emerald cut.
Marquise shapes were a very popular shape in the 1980s. Shaped like a football, the elongated shape can make the stone appear larger than it weighs.
Hugely popular in the 1990s and early 2000s, the Princess cut is a square cut diamond with excellent marketing. A name like Princess cut is much better than the technical term, square modified brilliant. With four pointed corners, princess cuts are a little more likely to sustain damage over time than other shapes.
Pear shaped diamonds are used not only as center stones but as side stones and drops. Sometimes called a teardrop shape, when it is worn on the hand, typically it is worn point toward the nail.
Finally, the Asscher cut diamond. This is essentially a square emerald cut; however, it was originally cut by The Royal Asscher Diamond Company (then the Asscher Brothers of Holland). The shape was popular in the 1920s and had a resurgence in the early 2000s. Along with the emerald cut, Asschers are popular in Art Deco style pieces.
There are other shapes you might encounter in an appraisal-- single cut, trillion, and baguette, to name a few-- but the ten above and these short explanations should help when you read your appraisal. Of course, if you ever have a question, feel free to ask, I am always more than happy to answer.