What is the difference between gravitational force and gravitational field? I see two different formulas describing gravity:
$$F=\frac{GMm}{r^{2}}$$
$$g=\frac{F}{m}$$
But I don't understand the difference between gravity as a force and its field as a vector.
 A: Gravitational force depends on both the source mass and the test mass while the field is purely a property of the source mass.
For concreteness, consider the gravitational force acting on a person of mass $m$ due to the Earth having mass $M$. The gravitational force on the person is then given by \begin{equation}F=\frac{GMm}{r^{2}}\end{equation} (I am ignoring the direction now). It says that in general, gravitational force magnitude depends on the source mass, test mass and their separation.
But there is more fruitful way of seeing this: Suppose the person does not exist. Then the Earth will, due to its mass $M$, spread a gravitational field everywhere. The gravitational field strength is given by \begin{equation}g=\frac{GM}{r^{2}}\end{equation}
This vector field clearly depends only on the Earth; it measures how much force the Earth will exert on a unit mass, hence $g$ is sometimes called gravitational force per unit mass. If you now put the person at distance $r$ away from the centre of the Earth, this person will fall under the influence of the field and experience a force equal to $F=mg$, and this gives back the first formula.
