Physical interpretation of fields in QFT How to best interpret fields? How does it keep with the definition of a field as something which has a value(scalar or vector) at every point in space time?
If we can relate the above definition to fields in QFT,what does the "value" of the field physically signify.
For example:We know that the value of the temperature field at any point gives temperature.
I am at ease with this one :field is a well behaved function which when quantised gives a particle of specific momentum at a space time point.
 A: In quantum field theories, the fields are there to make locality and causality manifest. This fields are suitable linear combinations of creation and annihilation operators of one-particles states. They are not necessarily hermitian  , witch means (in QM) that they are not necessarily observables.
They are something as:
$$
\phi_{\alpha}(x)=\sum_{p,\sigma,n}A_{\alpha}^{p,\sigma,n}(x)a_{p,\sigma,n}+\sum_{p,\sigma,n}B_{\alpha}^{p,\sigma,n}(x)a^{\dagger}_{p,\sigma,n}
$$
where $\alpha$ carries a finite representation of the Lorentz symmetry group. They are constructed such that $[\phi_{\alpha}(x),\phi^{(\dagger)}_{\beta}(y)]=0$ for space-like intervals. This makes our life easy to construct interactions that respect casuality and locality. Only making terms as 
$$
V=\int d^4x\, \partial^{(m)}\phi(x)^n...\partial^{(k)}\psi(x)^l
$$
Or any other non-linearity in the Lagragian or Hamiltonian. We need to do so because all physical quantities are sensible to interactions via the time-ordering:
$$
\mathcal{T}\exp\left(\frac{i}{\hbar}\int_{t1}^{t2}V(t)dt\right)
$$
and, in general, the time-ordering of a product of fields $\mathcal{T}(\phi_{\alpha}\phi^{\dagger}_{\beta})$ would dependent on the referential frame if $[\phi_{\alpha}(x),\phi^{(\dagger)}_{\beta}(y)]\neq0$.
The point is: There is maybe others ways to approach to relativistic QM without fields, see this for example. 
Now, you may be questioning about fields that shows up in macroscopic scale like EM field. Turns out that massless particles couples with massive particles in a such a way (gauge interactions) that makes easy some types of multi-particle states to arise, namely the coherent states.
This states has a non-definite number of particles, and a well shaped field. You can think of that as the premier manifestation of the wave-particle duality. There is a field observable $\mathcal{O}(x)$, with expectation value $\langle\mathcal{O}(x)\rangle = \mathcal{O}_{\mathcal{cl}}(x)$, and a minimal standard deviation. The vacuum state is a coherent state for $\mathcal{O}_{\mathcal{cl}}(x)=0$ everywhere. Any systems, in principle, can achieve this states, but there should be interactions that realize physically such states. In special relativity this state are very natural, specially for massless particles.
In non-relativistic QM the number of particles are always conserved, and superposition of states with different number of particles are impossible, which implies the non-realization of coherent states. This is so because the presence of a central charge in the Lie algebra of the Galilean Group. Central charges are threats to symmetries. They do not necessarily spoil the symmetry, but can spoil if the super-selection rules are disobeyed. 
A: Starting with nonrelativistic fields, (and later incorporating quantum mechanics):
A field is an  object/ entity which, at each point in space and time has a value, so they are in effect, “functions” of space and time. A field also satisfies  an equation of motion, and they have a significance physically in the sense  that they can carry from energy one place to another  and are capable of  affecting physical processes.
Moving on to relativistic fields:
Relativistic fields fall into 2 groups, based around satisfying  an equation of motion of either class $0$ or $1$, (which are classes of equations, dependent on the nature of the function $Z$).
$$\frac  {d^2Z}{dt^2} – c^2 \frac  {d^2Z}{dx^2} = 0,\qquad  \mathrm{Equations\ of\ class\ } 0.$$
( Electromagnetic waves moving through vacuum comply with equations of class  0, and travel at c).
or a equation of motion
$$ \frac  {d^2Z}{dt^2} – c^2 \frac  {d^2Z}{dx^2} =  – (2\pi  \nu_{\mathrm{min}} )^2 (Z-Z_0) \qquad \mathrm{Equations\ of\  class\ }1.$$    
$\nu_{\mathrm{min}}$ is the minimum frequency for waves in this field.
If equation of class $1$  have solutions with   amplitude $A$, frequency $\nu$, wavelength $\lambda$ and equilibrium value $Z_0$, then the equation of motion requires that the frequency and wavelength be related to the quantity $ \nu_{\mathrm{min}}$ that appears in the equation by the formula:
$$\begin{align} \nu^2 &= \left(\frac{c}{\lambda}\right)^2+ (\nu_{\mathrm{min}})^2 \\
&= \left(\frac{c}{\lambda}\right)^2+ (\nu_{\mathrm{min}})^2 \end{align}$$
The minimum frequency for any wave is just $\nu_{\mathrm{min}}$ , and setting $\nu = \nu_{\mathrm{min}}$ (and thus $\lambda \rightarrow \infty$) corresponds  to a vertical line. 
It is  possible to obtain the similar class $1$ relation by just setting $\mu = \nu_{\mathrm{min}}$ to zero; obtaining the square root,  we  have $\nu = c/\lambda$, which is basically a straight line.
In this situation, $\nu_{\mathrm{min}}$ is zero; the field is capable of oscillations at any frequency.
Now we need to incorporate Q.M. by placing discrete values on amplitude $A$ and these values are proportional to the square root of $n$, a positive integer (or zero), which is the number of quanta of oscillation in the wave. The energy stored in the wave is:
$$E = \left(n+\frac{1}{2}\right) h \nu$$
where $h$ is Planck’s constant. The energy associated with each quantum of oscillation depends only on the frequency of oscillation of the wave, and equals
$$\begin{align} E &= h \nu,\  (\mathrm{for\ each\ additional\ quantum\ of\ oscillation}). \\
E^2 &= (h\nu)^2 \\
&= \left(\frac{hc}{\lambda}\right)^2 + (h \nu_{\mathrm{min}})^2 \end{align}$$
Einstein’s theory of relativity gives us the relativistic energy equation. 
$$E^2 = (pc)^2 + (mc^2)^2$$
The minimum energy that object can have is just $mc^2$, (it's  rest energy)  which reinforces  the statement that the minimum frequency a class 1 wave can have is $\nu_{\mathrm{min}}$. This leads to the conclusion that, for a quantum of a relativistic field,
$$\begin{align} p c & = \frac{hc}{\lambda},\ \mathrm{and} \\
 m c^2 & = h \nu_{\mathrm{min}}, \end{align}$$
the familiar Einstein De Broglie  relations.
Class $0$ relativistic fields include electric fields and their waves are electromagnetic waves. The version of the formula above that we get for class $0$ quanta is the same as for class $1$ fields with $\mu = \nu_{\mathrm{min}} $ set equal to zero — in other words, with $m = 0$. 
The square root is:
$$E = p c$$
which is Einstein’s relation for massless particles. The quanta of electromagnetic waves  are indeed, once we apply the two equations above, massless particle, i. e.  photons.
From the second equation above, we can finally see what the mass of a particle is. Each particle that has a mass is a quantum of a class $1$ field whose waves have a minimum frequency $ν_{min}$; the minimum energy of a single quantum of these waves is $h$ by frequency,  and the particle mass  is that minimum energy divided by $c^2$.
$$m = \frac{h \nu_{\mathrm{min}} }{c^2}$$
To discover the source of the  particle’s mass , we need to learn what determines $\nu_{\mathrm{min}}$, and why a minimum frequency exists.  These are still open questions.
Particles are quanta of relativistic quantum fields. Massless particles  are quanta of waves in fields that satisfy a class $0$ equation.  Massive particles  relate to fields with a class $1$ equation. 
My thanks to @flippiefanus for pointing out that the remarks above refer to  bosonic fields.
Fermionic Fields
The Weyl equation deals with massless spin $\frac {1}{2} $ particles can be written: 
${\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0}$
explicitly in SI units:
${\displaystyle I_{2}{\frac {1}{c}}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0}$
where
${\displaystyle \sigma ^{\mu }=(\sigma ^{0},\sigma ^{1},\sigma ^{2},\sigma ^{3})=(I_{2},\sigma _{x},\sigma _{y},\sigma _{z})}$
is a vector whose components are the 2 × 2 identity matrix for $\mu$ = 0 and the Pauli matrices for $\mu$ = 1,2,3, and $\psi $ is the wavefunction - one of the Weyl spinors.
Most embarrassingly of all, I have left out the the Dirac equation (for massive  spin $\frac {1}{2} $ particles, which,  in the form originally proposed by Dirac is:
${\displaystyle \left(\beta mc^{2}+c\left(\sum _{n{\mathop {=}}1}^{3}\alpha _{n}p_{n}\right)\right)\psi (x,t)=i\hbar {\frac {\partial \psi (x,t)}{\partial t}}}
$
where $\psi = \psi(x, t)$ is the wave function for the electron of rest mass $m$ with spacetime coordinates $x$, $t$. The $p_1$, $p_2$, $p_3$ are the components of the momentum, understood to be the momentum operator in the Schrödinger equation. 
Using $\gamma $ matrices, the Dirac equation can be reduced to:
${\displaystyle i\hbar \gamma ^{\mu }\partial _{\mu }\psi -mc\psi =0}$
This answer is based on this website, Matt Strassler - Fields and my notes based upon other pages on the same excellent site. (Which includes illustrations which  I assume are copyright). Extracts from Wikipedia Weyl Equation  and Dirac Equation are also included.
