What exactly is a suface dipole moment density?

A surface dipole moment density describes how many point-like dipoles per unit area are located on a surface (two dimensional object).

The simplest electric dipole is composed of two point charges, one positive point charge +q and one negative point charge -q, that is separated by a distance d. The total charge of this system is zero which implies that, far away from this system, the lowest order E-field due to this system is given by the electric dipole moment (a vector quantity denoted by \vec{p} in Jackson notation):
\vec{p} = \int\ \rho(\vec{r}\,') \vec{r}\,'\ d\tau' = \sum_k q_k \vec{r}_k
where \rho is the volume charge density, d\tau' is the differential volume element in Griffiths notation, \vec{r}\,' or \vec{r}_k is the location of one of the charges that make up the system. q_k is the charge of the k-th point charge for a set of discrete charges.

If our electric dipole is centered on the x-axis, then its electric dipole moment would be:
\vec{p} = (-q)(-d/2)\hat{x} + (+q)(+d/2)\hat{x} = q d \hat{x} = p \hat{x}

A point-like electric dipole is an electric dipole in the limit where the separation distance goes to zero d \rightarrow 0 such that the magnitude of the electric dipole moment remains constant, which implies that q \rightarrow \infty as a part of this limiting process.

The units of the surface dipole moment density (D in Jackson notation) is charge per unit length, which can be more easily seen in the following way:
\frac{\text{(number of point-like dipoles)(average dipole moment magnitude)}}{\text{unit area}} = \frac{\text{(unitless)(charge)(distance)}}{\text{area}} = \frac{\text{charge}}{\text{length}}

An alternative way to describe a system of point-like dipole smeared over a surface is the following. Imagine one charge layer (all positive) with a uniform surface charge density +\sigma (charge per unit area). Now imagine another charge layer (all negative this time) also with a uniform (and equal in magnitude) surface charge density -\sigma. These two charge layers are separated by some non zero distance d.

The surface-dipole moment density D is the mathematical idealization that exists in the limit that the separation distance d between these two charge layers goes to zero in such a way that the product of the separation distance d with the surface charge density \sigma remains a constant in this limit. This limiting constant is D = \lim_{d \rightarrow 0} \sigma d.

An interesting consequence of a surface dipole moment density is the discontinuity of the electric potential that occurs as one traverses from one side of the surface to the other side. The fact that a discontinuity exists in the electric potential seems to be conflict with the general idea that the electric potential is continuous everywhere and we use that boundary condition all the time to connect the potential at the boundary between two different materials.

These two perspectives can be reconciled by the experimental observation that point-like electric dipoles do not seem to exist. If point-like electric dipoles were to exist, then their existence would imply the existence of non-electromagnetic forces between subatomic particles that are not the same when the arrow of time is reversed (time-reversal violation). The search for these exotic objects are a topic of contemporary research including at FRIB which will produce rare pear-shaped nuclei that have enhance the observability of these objects by a factor of 1000x or more.

Setting aside the issue of whether point-like electric dipoles exist, why would a surface smeared with a distribution of dipoles result in a discontinuity in the electric potential?

Let’s imagine again two surface charges densities with equal in magnitude but opposite in sign charges separated by a distance of d. If we were to zoom in on a small patch of these surfaces, then they would, to a very good approximation, resemble a parallel plate capacitor. Above and below but outside the capacitor plates, the E-field is zero. Within the capacitor plates, the E-field is a constant (within this approximate picture).

Assuming the two plates of the capacitor are parallel to the xy-plane and are separated by a distance d along the z-axis, the potential difference across the capacitor can be given by the line integral:
\int_A^B \vec{E}\cdot d\vec{\ell} = V_A-V_B
Outside the capacitor plates, there is no E-field, so this region does not contribute to the line integral. The only non-zero contribution to the line integral comes from the region inside the capacitor. Let’s assume that point A is above and outside the capacitor and B is below and outside the capacitor. In this case the line integral gives:
\int_A^B \vec{E}\cdot d\vec{\ell} = \int_0^d E (\hat{z} \cdot \hat{z})\ dz = dE = V_A-V_B = \Delta V
where we have assumed that the positive plate is at z=0, the negative plate is at z=d, and the E-field therefore points up the positive z-axis.

In between the two plates of capacitor, the E-field contribution from the positive plate is +(\sigma/2\epsilon_0)\hat{z} and the E-field contribution from the negative plate is +(\sigma/2\epsilon_0)\hat{z}. This implies that E = \sigma/\epsilon_0 in between the plates, where |\sigma| is the magnitude of the surface charge density on either plate.

We can plug this into our calculated value for the potential difference across the plates to get \Delta V = dE = d (\sigma/\epsilon_0) = (d \sigma) /\epsilon_0. If we now perform the limiting process for the separation distance as described before d \rightarrow 0, then the surface point-like dipole moment density is D = \lim_{d \rightarrow 0} \sigma d.

Putting this altogether, we find that the potential difference is \Delta V = D/\epsilon_0.