single- and double-layer potentials
If a point charge gives the basic 1/r potential, what does a charged surface give? Spread charge in a thin sheet over a boundary and you get a single-layer potential — the superposition of the point-source potentials of all the charge on the sheet. Pair up positive and negative charges into tiny dipoles all over the surface and you get a double-layer potential. These two surface potentials are the raw material for solving boundary-value problems by recasting them as equations on the boundary alone.
Precisely, both are integrals of the fundamental solution over a surface S. The single-layer potential with density mu places charge of density mu on S and integrates the kernel (1/r in 3D, log r in 2D) against it; the result is continuous across S, but its normal derivative jumps by an amount equal to the density — that jump is the surface charge made visible as a discontinuity in the field. The double-layer potential with density nu integrates the normal derivative of the kernel (a dipole at each point) against nu; this one is itself discontinuous across S, jumping by an amount proportional to the density as you cross the surface. Both are harmonic off the surface, by construction.
Their purpose is the boundary integral method, a beautiful reduction. To solve the Dirichlet problem you guess that the solution is a double-layer potential of some unknown surface density and impose the boundary condition; the jump relations turn this into an integral equation for the density on the boundary surface — a problem in one fewer dimension. The Fredholm theory of such integral equations then delivers existence and uniqueness, and numerically the boundary element method discretizes exactly these surface integrals, meshing only the boundary rather than the whole interior. The honest subtlety is precisely the jump and limit behaviour: the layer potentials are singular and discontinuous right on the surface, and getting their boundary limits right is the whole art of the method.
A uniformly charged spherical shell produces a single-layer potential that is constant inside the shell (no field there) and falls off like 1/r outside, with the field jumping discontinuously across the shell by exactly the surface charge density — the textbook electrostatics result, read directly off the single-layer jump relation.
A surface of sources; single-layer jumps in flux, double-layer jumps in value.
Remember which jumps: the single-layer potential is continuous but its normal derivative jumps; the double-layer potential itself jumps in value. Mixing these up wrecks the boundary integral equation. Both are harmonic only OFF the surface — on the surface they are singular and must be handled by their limit relations.