The sparkle of a diamond is a product of the cut, and the three parts of sparkle (brilliance, fire, and scintillation) are all determinates of cut quality from the appearance of the diamond.
Brilliance + Fire + Scintillation = Sparkle
We know that the highest quality diamonds are the ones with the best cut. Cut is a crucial factor for a diamond because it determines the polish and symmetry of the diamond when cut from rough material. And, taking a step backwards, the cut is determined by the craftsmanship of the diamond cutter. The angles of the cut must be perfectly aligned so that light is reflected and refracted.
Scintillation: (noun) "a flash or sparkle of light produced in a transparent material"
Scintillation is essentially the part of the sparkle that happens when the diamond moves, according to the GIA. Ideally, the gemstone evenly refracts light across its surface with few dead spots. The light the diamond is in determines the balance of brilliance and fire in the sparkle.
Brilliance: (noun) "intense brightness of light"
The brightness and contrast of white, pure light determines the brilliance of a diamond when it is face-up to the observer. Diamonds are like mirrors when reflecting light, so if the diamond is cut too deep or too shallow, less light will be reflected through the top of the diamond. However, the contrast of the light is important as well, because the opposition of light and dark actually makes the light look brighter.
Fire: (noun) "flashes of rainbow colors"
Fire occurs when diamonds exhibit color. White light traveling through the diamond is dispersed into a spectrum of colors. High quality diamonds are usually basically colorless (see Color), but a rainbow of colors is still refracted. More fire is produced when the cut has steep crown angles and a smaller table. Unfortunately, this means less white light is reflected, so it is important to find a balance of the two. Old cuts and fancy cuts generally display more fire because of the steep crown angles and small table. Additionally, fire is swamped by strong brilliance, or bright white light, so fire usually shows more in darker environments, such as a restaurant.