What are fields? I'm following my first course in field theory and the professor began, like many books do, by introducing the scalar field. However, I am a bit hesitant about the physical idea of fields. My question is: what is the physical meaning of the fields? Why they are introduced?
I read the introduction of the books of Peskin and Weinberg but I'm not satisfied.
 A: Fields are important in both classical and quantum physics.
At first it was thought they are just some imaginary mapping of the forces of gravitation or electric and magnetic forces. Gradually it became clear one can not keep it away from physics. They are more than imagination.
Why? It is because electromagnetic effects take some finite nonzero time to reach from one point to other. A moving charge takes a nonzero time to influence another charge or a magnetic needle. If we want to preserve the laws of conservation of momentum and energy during that nonzero time we must have to take the concept of field as real and it should be thought of as the seat of energy and momentum.
At first people tried to understand field in terms of classical mechanics. It was believed at first that there must be some mechanical medium whose stress and strain must be the manifestation of field. The medium was named as aether. But all such efforts failed.
It became clear that field is a fundamental entity of nature just as matter is. Or maybe matter is itself a kind of concentrated field!
Quantum theory has given rise to quantum field theories and the classical fields have been quantized in QFT. It is now understood that fields can be thought of as consist of field quantas or particles with integer spin (bosons) and the so called matters are particles of half interger spin (fermions).
A: Probably the most fundamental and simple idea of the field arises from heat equation. You have a heat source and heat diffusion through media. It is described by field of temperature. It is the simplest scalar field I can imagine. But it has nontrivial equation of motion -  has it? From that simplest cases more complicated arises: force fields introduced without any complicated mathematics by well known genius Michael Faraday, who just draw lines of equal potential from one charge to another in order to understand how electrostatic and magnetic forces works. Of course You may imagine even more complicated fields - general tensor or spinor ones. 
Strict definition of fields is of course a function which for given point on the manifold assign tensor or spinor to it. Physical fields often has strong continuity and differential conditions on them and obeys complicated differential or integral equations, but the most fundamental idea is still the same: lines which was drawn by Faraday, temperature in continuous media, velocity field of the fluid passing through water canal. Try to understand that simple ideas and return to "quantum-compacted-26-dimensional-theories of everything".
A: I am a layperson who knows little about science but am eager to learn. Perhaps  a field  can be described as an invisible medium  that enables action between objects and  the effects of which can be measured. 
Brian Silver, The Ascent  of  Science,  gives basic and more advanced  descriptions of this concept. 
