In materials science, hardness is the characteristic of a solid material expressing its resistance to penetration, and by extension its power of penetration in softer materials. An object made of a hard material will scratch an object made of a softer material.
Hardness is usually measured on various scales, such as the Rockwell scale or the Mohs scale.
Pure diamond is the hardest known substance (on every scale), so it will scratch any other material. Diamond, being the hardest material, is the substance used to cut diamond. Higher-grade diamonds are used to cut lower-grade diamonds.