The hardness of a mineral is its resistance to being scratched. If you strike a mineral with a hammer and it breaks, leaving surfaces that are rough and uneven, it is said to fracture.
Cleavage is the way that a mineral breaks along well-defined planes of weakness.
Often these planes are between layers of atoms or other places where the atomic bonding is weakest. Cleavage surfaces are not perfectly smooth. One of them is cleaving. Most preliminary shaping of diamonds is done by sawing, which may be done with a thin, rapidly revolving phosphor-bronze disk. The edge of the saw is impregnated with diamond powder.