Does electric field carry charges? A charged particle say an electron carries a negative charge and it will experience a force when there is an electric field, like charges repel and unlike charges attract so does this means an electric field either carry charge(s) or behave like charge, or charge can only be applied to particle? Also can electric field be neutral?
 A: Yes, indeed fields can carry charge, however the EM field (also called the photon field) does not carry any charge. If it did, then EM fields could affect the trajectories of photons. 
I will do my best to explain why below, but there's no way to do it without group theory (as far as I know).
The meaning of carrying a charge means that it transforms under some non-trivial representation of the gauge group. So, for instance in EM the fermions $\Psi$ transform non-trivially under the gauge group $U(1)_{EM}$, however the EM field $A_\mu(x)$ transforms under the trivial representation of $U(1)_{EM}$ (meaning it doesn't transform at all), however this is in particular a special case, since $U(1)$ is an abelian group. 
However, fields can carry a charge. In the strong force the fermions $\Psi$ carry a different kind of charge: They carry an $SU(3)_c$ color charge, and they transform under the fundamental representation of $SU(3)_c$ (i.e. they have 3 color indices). Moreover, since  $SU(3)_c$ is a non-abelian group it turns out that in order for this to be a symmetry of our lagrangian, the gauge fields $A_\mu^a$ do transform under the gauge group, meaning that they indeed do carry a color charge as well. However, they are in what is called the adjoint representation of $SU(3)$, and so carry 8-color indices (which is what the $a$ in $A_\mu^a$ represents) . 
