How does gravity affect an object?

Gravity imparts a force on an object. That force is proportional to the mass of the object. That force can cause the object to move, i.e. to accelerate and change velocity and/or distance. Other than that, gravity does not affect an object.

f = ma

where f = force in newtons, m is mass in kilograms, a is acceleration in meters per second squared. That suffices for the one body problem with a known gravity field, such as a=9.81 m s-2 on the Earth.

For the two body problem, such as Earth/Moon calculations or where a is unknown, you can use

f = G(Mm/r2)

where f = force in newtons, G is the universal gravitation constant, 6.674 x 10-11 n m2 kg-2, M and m are the masses of the two bodies in kilograms, and r is the distance between them in meters.