Step 1: Concept The Brinell hardness test measures the resistance of a material to permanent indentation by a hardened steel or carbide ball.
Step 2: Meaning The Brinell Hardness Number (BHN) is defined as the ratio of the applied load to the surface area of the resulting indentation.
Step 3: Analysis Load is typically measured in kilograms-force ($kgf$) and the surface area of the spherical cap indentation is measured in square millimetres ($mm^{2}$).
Step 4: Conclusion Thus, the unit of BHN is $kg/mm^{2}$.
Final Answer: (D)