Electric Displacement Vector How do I interpret what electric displacement vector is? I know that it exists and I know it's an equation but I'm not able to really understand or interpret what it is.           
$$\oint_A \mathbf{D} \cdot \mathrm{d}\mathbf{A} = Q_\text{free}$$
 A: Well think that in vacuum, the electric field vector $\bf E$ is a good measure of the strength of the electrostatic field. 
But if you are measuring inside a medium, say there are some charges around where you are measuring, now the presence of those charges affects the value of $\bf E$ from the external electric field. These charges will arrange themselves differently (depending on how free they are to move in the medium) and their configuration will change the electric field from $\bf E$ to a new value $\bf D.$
Also the expression you write, which is a special case of Gauss' Law, is telling you that the difference in electrical field intensity between the inside and the outside of the medium, is linked to the charges concentrated on the surface. Indeed, since this integration is independent on the chosen surface $\bf A,$ you can chose a cylinder whose axis is normal to the medium's surface, and with one face inside and the other outside. Then the integration will equate to the integral of charge populating the surface.
A: I have found several question on the meaning of displacement vector $\vec{D}$, how it is different from the electric field $\vec{E}$, what is the physical meaning of $\vec{D}$ etc on physics stack exchange. The reason for asking so many questions on displacement vector $\vec{D}$ (as I guess) perhaps comes due to the fact that most of the people are following a top down approach on electrostatics. i.e. they stumbled upon $\vec{D}$ and try to understand about it. However it will be more thrilling here to follow a bottom up approach i.e. to understand why it was necessary to introduce $\vec{D}$ and how the concept of $\vec{D}$ could survive the robust logical formulation and framework of fundamental physics.
Like most of fundamental physics, Coulomb's law is an experimental law which is mathematically written as:
${\nabla}\cdot\vec{E}=\rho/\epsilon_0$
Here both $\vec{E}$ and $\rho$ are measurable quantities. Coulomb's law tells the following: Just take any region of empty space (containing some charge in it) and "measure" the electric field at every point in this region of space (please note that electric field is an experimentally measurable quantity; just put a test charge at a point and measure its accleration then apply E=F/q). Then calculate the divergence of the electric fields in this region from your measured data. The result, Coulomb's law says, is $\rho/\epsilon_0$. You can cross check it by measuring $\rho$ by a different experiment (just connect the charge distrubution through an ameter to ground and measure the current and duration of flow of this current. Then you can find the charge(Q=Ixt) and easily calculate the $\rho$. Thus a physical law is a relation between "measurable" quantities. If you know $\vec{E}$ then you know $\rho$ unambiguously and vice versa.
Now consider the case of dielectric slab sandwiched in a parallel plate capacitor as shown in the figure below.

If you measure the electric field $\vec{E}$ inside the dielectric slab, then you get a difference of $\vec{E_0}-\vec{P}$. A difference is not unique. For example 10-5=5 and 1005-1000=5. So $\vec{E}$ can not tell about any charge distribution in the system. This can not be a quantity that satisfies Coulomb's law. So $\vec{E}$ is a meaningless quantity in the context of a dielectric.
Now, we can measure the free charge in the capacitor by discharging it through an ameter for example. This is an unique quantity of the system and we have to find another measurable quantity of the system so that we can relate it to the free charge density. By writing $\nabla\cdot\epsilon_{0}\vec{E}=\rho_{free}+\rho_{bound}$ we arrive at the relation $\nabla\cdot\vec{D}=\rho_{free}$ (Coulomb's law like rule) where $\vec{D}=\epsilon_0\vec{E}+\vec{P}$. Thus $\vec{D}$ satisfies Coulomb's law like rule with the free charge density inside the dielectric.
It should be noted that $\vec{D}$ is a measurable quantity. Just take any dielectric slab. Inject some free charge into it. Then all over points on the surface of the delectric measure the charge/unit area.That is $\vec{D}$ at that point. Then calculate the divergence of $\vec{D}$ from the measured data. The result is $\rho_{free}$ which is again a measurable quantity.
Thus $\vec{D}$ is not same as $\vec{E}$ They are separate quantities and also they are dimensionally different. But $\vec{D}$ is significant because $\vec{D}$ connects to free charge density($\rho_{free}$) in dielectric materials by obeying a Coulomb's law like rule just as $\vec{E}$ connects to $\rho$ in vacuum. This quality of $\vec{D}$ makes it possible to write Maxwell equations for dielectrics. This was the necessity of the new quantity D.
