weight force is the force(gravitational) of the earth on an object and the normal force is also a force by the earth(surface) on an object. Action reaction pairs must point in opposite directions, that is true, but they must also act on DIFFERENT bodies, therefore weight force and normal force are not an action reaction pair, they both act on the object...
Gravitational mass is

, which is a re-arranged version of newtons law of gravitation. In other words, it is the mass that exerts/experiences a certain force due to the attraction between another mass(

) at a distanct

.
Inertial mass is

, which is essentially the how hard something is to accelerate when a force is applied, larger mass means harder to accelerate, i.e. lower acceleration...
It turns out that both m's have the same value to some crazy accuracy value that is really the precision of the instruments used, but apparently there is no reason that they should have the same value, it just turns out that they do(according to my physics lecturer).