1) Inertial mass
This is defined by Newton's 2nd law- F = ma, which states that when a force F is applied to an object, it will accelerate proportionally, and that constant of proportion is the mass of that object. In very concrete terms, to determine the inertial mass, you apply a force of F Newtons to an object, measure the acceleration in m/s2, and F/a will give you the inertial mass m in kilograms.
2) Gravitational mass
This is defined by the force of gravitation, which states that there is a gravitational force between any pair of objects, which is given by
where G is the universal gravitational constant, and are the masses of the two objects, and r is the distance between them. This, in effect defines the gravitational mass of an object.
Gravitational mass is measured by comparing the force of gravity of an unknown mass to the force of gravity of a known mass. This is typically done by balance scale.