free-space Green's function
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Before worrying about walls, ask the simplest version of the question: how does a point source spread its influence through an infinite, empty medium? The free-space Green's function is the answer — the Green's function of an operator on all of space (or space-time), with the only condition being that the field decay sensibly far away rather than match any particular boundary.
It is essentially the fundamental solution, viewed as a function of source point s and field point x and, crucially, depending only on their separation x - s when the operator is translation-invariant. For -nabla^2 in three dimensions it is G_free(x, s) = 1/(4 pi |x - s|); in two dimensions G_free = -(1/(2 pi)) ln|x - s|; for the heat operator it is the Gaussian kernel (4 pi t)^{-n/2} exp(-|x - s|^2/(4 t)). Because it depends only on x - s, applying it to a source is a convolution: u(x) = integral of G_free(x - s) f(s) ds, which Fourier transforms turn into simple multiplication.
The free-space Green's function is the reusable atom of boundary-value theory. To solve a problem in a bounded or half-space region you start from G_free and add a homogeneous correction — most cleanly through the method of images — so the sum satisfies the actual boundary conditions while still producing the same point source. It is also the object that appears directly whenever the geometry really is unbounded: radiation from an antenna into open space, scattering of waves, or the potential of an isolated charge distribution.
The potential of a point charge q at the origin in free space is phi(x) = q/(4 pi epsilon_0 |x|), which is q/epsilon_0 times the free-space Green's function of -nabla^2. A continuous charge density rho gives phi(x) = integral of rho(s)/(4 pi epsilon_0 |x - s|) ds — superposition of point potentials.
Coulomb's potential is the free-space Green's function of the Laplacian, dressed with physical constants.
The free-space Green's function alone does not solve a bounded problem — it satisfies the equation and the point source but not the wall conditions. Using it as if it did is a common error; you must add the homogeneous correction that the boundaries demand.