Abrasive
An abrasive works by being harder than what it is cutting, and the ranking is therefore almost the hardness ranking — with two qualifications that matter. Toughness decides whether a grain fractures usefully or crumbles, and chemical compatibility decides whether it survives contact at temperature: diamond is the hardest thing there is and is the wrong abrasive for steel, because carbon dissolves into hot iron.
The other axis is whether the grain is meant to survive. A friable abrasive breaks to expose fresh edges and stays sharp as it wears away; a tough one holds its shape and eventually goes blunt. Which is wanted depends on the machine, not on the material being cut.
Compound
Aluminium oxide
Industrial alumina — the refined intermediate between bauxite and aluminium metal, and a major ceramic in its own right.
Mineral
Corundum
Aluminium oxide — hardness 9, the mineral of ruby and sapphire, and a major industrial abrasive.
Material
Sand
historically in sandblasting, now largely replaced by garnet and slag because breathing crystalline silica causes silicosis
Compound
Silicon carbide
A synthetic ceramic close to diamond in hardness, and the first commercially successful manufactured abrasive.
Compound
Tungsten carbide
and more importantly a cutting material — carbide holds an edge at temperatures that would anneal any steel
Each of these uses is established and cited on its own. Whether something counts as an ore, for instance, depends on grade, price and era — and those qualifications are shown alongside it.