What is the physical significance of polarization vector? What is the physical significance of the polarization vector in studying dielectric materials? Why do we need that term, what information does it give about the property of the dielectric?
 A: The polarization vector $\mathbf P$ is the volume density of dipole moments in a dielectric material. In other words 
$$\mathbf P=\frac{d\mathbf p}{dV}$$
Where $\mathbf p$ is the dipole moment and $V$ is volume (this is similar to saying mass density is $\rho=\frac{dm}{dV}$).
This polarization vector relates specifically to "bound charges" in our material through 
$$\rho_b=\mathbf\nabla\cdot\mathbf P$$
where $\rho_b$ is the bound charge density. This is a sort of "Gauss's law" for bound charges in a material.
The reason this term is so important is because it helps with studying electric fields in matter. This is done by defining a "displacement field" $\mathbf D=\epsilon_0\mathbf E+\mathbf P$. It turns out that we can get to the following relationship $$\rho_f=\mathbf\nabla\cdot\mathbf D$$ 
where $\rho_f$ is the free charge density in the material.
So as you can see, the polarization vector allows us to look at the effects of free and bound charges separately when looking at the overall electric field $\mathbf E$.
In terms of what information it gives us about the dielectric, it gives us no information in general. $\mathbf P$ depends on what other fields are present. You could achieve the same $\mathbf P$ in different dielectrics. If you put specifications on what system the dielectric is in, then maybe you could get something, but usually it works the other way around. For example, if we say we have a "linear dielectric" then we can say that $\mathbf P=\chi\epsilon_0 \mathbf E$, so that the properties of the dielectric can determine how $\mathbf P$ behaves.
