Why are fields described as force divided by mass or charge? I have read that application of force on a body from a distance, like gravitational or electrostatic force is a two-step process, first, the field is created by the body, then, the application of force on the second body by the field. I want to know why the expression for gravitational field is given as F/m or why the expression for electric field is given as F/q?
 A: 
Why are fields described as force divided by mass or charge?

Because they follow from the classical universal law of gravitation  and Coulomb's law.
The force that each of two masses or charges experience is due to the gravitational or electric field generated by the other mass or charge.
For gravity the magnitude of the force, where the centers of the masses are separated by distance $r$, and $G$ is the universal gravitational constant, is given by the universal law of gravitation
$$F=G\frac{m_{1}m_{2}}{r^2}$$
So if one is interested in the magnitude of the gravitational field $g$ due to $m_2$ that causes a gravitational force $F$ on $m_1$, it is given by $F/m_1$ or
$$g=G\frac{m_2}{r^2}$$
Similarly, the electrical force between two point charges is given by Coulombs law where $k$ is the Coulomb constant
$$F=k\frac{q_{1}q_2}{r^2}$$
And the electric field E due to $q_2$ that causes a force $F$ on $q_1$ is
$$E=k\frac{q_2}{r^2}$$
Hope this helps.
A: The definition of field, is there to tell us about the effects of the field on an object of unity value. most force fields have the parameter of the object they effect as a multiplier, hence when you set the parameter to unity value, it is the same as dividing the force by that parameter.
A: Let me start by eliminating a possible misunderstanding. You wrote

...application of force on a body from a distance, like gravitational or electrostatic force is a two-step process, first, the field is created by the body, then, the application of force on the second body by the field...

It is ok, but it should be clear that the two steps are just conceptual steps. There is no cause-effect or time relationship between the two steps.
Now, about your statement fields are described as force divided by mass or charge, it is not always true. In general, the force between two bodies is something depending on both in a more complex way. Only in special cases, it may be considered as the product of a quantity depending on one body (the field at position of the second body) times an intrinsic property of the other body (its mass or charge). That is certainly the case of two point-like charges or two point-like masses. But it is the limiting case of a more complex situation, in general.
Let's consider, for example, the case of a planet and its satellite. The real interaction is not simply given by the Newton formula
$$
F = G \frac{m_1 m_2}{r^2}
$$
at all distances, because we have extended bodies and tidal forces deform them. The real interaction has to take such effects into account in a self-consistent way, resulting in an angular-dependent force. Similarly for the interaction between two charged spheres. Depending on whether they are insulators or conductors, one has to take into account different charge redistribution on each sphere due to the presence of the other.
In such cases, one can still introduce the concept of the field due to only one body as the force per unit mass or unit charge, in the limiting case of a very small mass or charge. More formally, the general definition of the gravitational field or the electric field created by a body is
$$
E = \lim_{p \rightarrow 0} \frac{F}{p},
$$
where $F$ is the intensity of the force on the test body, and $p$ is the mass of the test body in the gravitational case or the charge in the electric case. The meaning of the limit here should be intended as considering values of the property $p$so small that there is a negligible modification of the properties of the first body.
Now, let's go to the why. It is clear that the whole construction of the concept of field has the aim of factorizing in an exact or approximate way the interaction between bodies into a part (the field), depending only on the properties of the source, and another depending on the property of the other bodies. Such a factorization is helpful to simplify many problems, although, in general, should be considered only as an approximation.
However, there is something more. The concept of the field remains auxiliary until we introduce the possibility that the field itself becomes a new dynamical quantity. At this stage, we have to introduce some equation of motion for the field. The intensity of the force on a test particle remains $p E$, provided the presence of the test particle does not perturb the field too much.
A: Answering my question for anyone who benefits from this.
This is what I understood of everything I read about it . Electric field is simply like a constant of force applied by a particle at a particular point in space, a ratio which is made independent of the test charge by dividing the force formulae by the mass/charge of the test particle, since the magnitude of force contains the charge as a multiplication term and is directly proportional to it. Although it's an entirely new physical entity that exists whether or not the test particle is present.
I am reading a paper which further goes into the history of electric fields and other fields.
It's "Introducing electric fields" by John Roche, 2016.
A lecture by Prof. H. C. Verma in Hindi also helped me understand it.(Available on YouTube in a playlist called Classical Electromagnetism)
