What is meant by polarization of an Electric Field? What is meant by polarization of an Electric Field? Say an Electric Field is polarized in the $x$-direction but is moving in the $z$-direction.
Can anyone please give some level of abstraction to this concept?
 A: An electromagnetic wave can oscillate in any direction perpendicular to the direction of the wave propagation. This direction of oscillation is called the polarization of the wave. 
In your example, the wave is moving in the $z$ direction, so it can oscillate in any direction in the $x-y$ plane. 
If the wave was polarized in the $y$ direction, it's equation would be: 
$$E=E_o \cos(\omega t-kz)\hat{y}$$
This is called linear polarization.
Two linearly polarized waves (one in the $x$ and one in the $y$ direction and out of phase by $\pi/2$) can combine to form a circularly polarized wave. 
This wikipedia article has beautiful gifs related to the topic.
A: Suppose the electromagnetic wave is propagating in the $z$ direction. Imagine placing a large plate parallel to the $xy$ plane (at $z=$ some constant). The electric field of the propagating wave continually hits the plate at different points lying on the plate$^{[1]}$. The locus of those points will define the polarization:
1) Linear: the locus is a straight line (could be of any orientation in the $xy$ plane)
2) Circular: the locus is a circle
3) Elliptic: the locus is an ellipse
For linear polarization, it can be a single wave or a superposition of waves. But when it's a superposition, the superposing waves' electric fields must all have the same phase.
To produce a circularly/elliptically polarized wave, there must be a superposition. Suppose we have 2 such waves (of equal amplitudes) superposing to give the final wave (the wave which we observe on the plate). If the phases of the electric fields of the 2 waves differ by $\frac{\pi}{2}$, we get a circularly polarized wave. That is, the resultant electric field vector has a constant magnitude, but is moving in a circle when you project it onto the plate. If the phase difference is something other than $\frac{\pi}{2}$ (or if amplitudes are different), the resultant wave is elliptically polarized.
It is a good idea to visualize this concept. There are a couple of animations illustrating that. One of them is this: https://youtu.be/Fu-aYnRkUgg
[1]: If this is still unclear, visualize yourself shooting a water gun at a wall. Now shake your gun. The pattern of water you get on the wall will tell you the polarization.
