Mass is not a single idea. It has two aspects :
Gravitational Mass – the mass of a body because of it's gravitational interactions. Two bodies close to each other exert gravitational forces on each other given by
(1) . Each one exerts the same gravitational force on the other.
Inertial mass – the mass of a body that represents it's resistance to acceleration. This is the mass that appears in Newton's Second Law of Motion,
Fortunately the gravitational mass is equal to the inertial mass and mass is a well defined and well understood concept. If the two were not equal the acceleration due to gravity might well be different for different bodies, since from (1)
and Einstein's General Theory of Relativity, which assumes them to be the same, would not be possible.