While shape is the outward shape of the stone, the word cut is used to denote how well a diamond returns light. Remember physics class? "The angle of incidence is equal to the angle of reflection." Well, when a diamond is cut correctly, the light will go in from the top, bounce around and return out the top like the "Perfect" in the graphic. If the stone is too shallow or too deep, the light won't hit at the correct angle to create the brilliance a diamond should.
There are three variations of light return, all of which are important in different ways. Brilliance= Return of white light
Fire= Return of spectral light
Scintillation= Flashes of light when the diamond or viewer is moving
Laboratory Reports will give a cut grade on round brilliant (and some cushion and princess) cut diamonds. Ideal cut or Excellent cut diamonds will return the most light, be incredibly sparkly, and will carry a premium. The American Gem Society, or AGS, set the standard for grading cut with Ideal/0 as their top cut grade, and The Gemological Institute of America, GIA, uses Excellent as their top grade.
While an Ideal cut diamond will always grade Excellent, an Excellent cut might not always grade Ideal because AGS and GIA use different methodology regarding cut. AGS is purely mathematical-- if a diamond is cut within precise parameters of percentages and angles, etc, it will grade Ideal. GIA judges based on light return, so if a diamond returns nearly 100% of light to the eye, it is Excellent even if an angle is slightly outside the strict parameters.
For fancy shapes-- anything that isn't a round brilliant-- there are standards, but the most important is something called Shape Appeal. If the stone has a pleasing shape it will look better and will be more valuable. For example, a princess cut should be square instead of rectangular, or an oval should be a pleasing oval and not too long and thin. Of course, Shape Appeal and industry standards only mean so much-- if you love a shape, then it doesn't matter if it's not in demand. Your appraisal value will reflect the pricing of replacing with something identical regardless.
I could go on for hours as this is only the tip of the iceberg, but I will rest it here. As always, if you have questions, please don't hesitate to ask whether it's about an appraisal, diamonds, and jewelry in general or anything else!