Clarity is the summation of any inclusions or surface irregularities on the diamond. The more there are the lower the score. Some of these irregularities can only be seen under magnification. These will get a grading of SI or slightly Included. Above that you have VS and VVS which stands for very slightly and very very slightly with Internally Flawless being the best.
The best diamonds are colorless or devoid of yellow which is a grade of D,E or F. Near colorless or very little yellow grading starts at G and goes to J anything after J has more yellow and is less desirable.
Cut and shape should not be confused. The better the cut the more reflective the diamond is. A well cut diamond allows light to enter through the table or top and reflect inside before bouncing back out through the table. This will garner a rating of Ideal Cut or Premium. Poorly cut diamonds allow the light to enter through the table but allow light to escape from the sides and bottom. A rating of Fair & Poor will be given for obvious reasons. Diamond sizes vary but the bigger they are the more expensive they are. This is measured by Carats.