This has two meanings in relation to diamonds.
Symmetry - The specific quality of cut within the shape.
Shape - Square, oval and so on.
As the only characteristic of a diamond not influenced by nature, the cut is open to mistakes and bad practices. Cut a diamond incorrectly and the defining sparkle will be compromised. It is how the 57 or 58 facets (the tiny planes cut on the diamond's surface) are angled and sized that dictate how light reflects and exits the diamond, an effect known as its "fire". Make the cuts too deep or too shallow and the diamond will be less brilliant
As shown below, a diamond with too shallow a cut will lose light from the bottom.
One that is too deep will lose light from the sides
The well-cut diamond will reflect its light upwards in the form of sparkle and brilliance
There are many different shapes to choose from. Some of the more popular cuts are shown below