The weak equivalence principle is (from the Weinberg book)
At every space-time point in an arbitrary gravitational field it is possible to chose a 'locally inertial coordinate system' such that, within a sufficiently small region of the point in question the laws of motion of freely falling particles take the same form as in an unaccelerated coordinate system in the absence of gravitation.
Weinberg than says later on that the weak equivalence principle is just the restatement of the equivalence of inertial and gravitational mass. I don't understand this. He also makes an example. He considers an object that falls towards the Earth. And then he considers a coordinate system that moves with $x'=x-gt^2$. In this coordinate system the object that moves towards Earth is at rest. Weinberg says that this is because there is a cancellation between gravitational and inertial forces!? I cannot see this explicitly...
The gravitational force between two gravitational masses $m$ and $M$ is $F_g=G\,m_g\,M_g/r^2$.