a fundamental solution

Imagine you want to know how a whole pond responds to a single pebble dropped at one spot. If you knew the ripple from one pebble exactly, you could predict the surface for any pattern of pebbles by adding up the ripples. A fundamental solution is that one-pebble answer for a linear differential operator: the response of the operator to a single, infinitely concentrated point source. Once you have it, you can build the response to any source by adding (integrating) shifted copies of it.

Precisely, for a linear constant-coefficient operator L (think of the Laplacian, the heat operator u_t - k u_xx, or the wave operator u_tt - c^2 u_xx), a fundamental solution E is a distribution satisfying L E = delta, where delta is the Dirac delta — the idealized point source of unit strength. Because L has constant coefficients, the answer to a source placed at a point y is just E shifted to y, namely E(x - y). Superposition then says the solution of L u = f for a general source f is u(x) = integral of E(x - y) f(y) dy, a convolution: you smear the point-response across the source. For the Laplacian in three dimensions the fundamental solution is the Newtonian potential -1/(4 pi |x|); in two dimensions it is the logarithmic potential; for the heat operator it is the heat kernel; for the wave operator it is the retarded fundamental solution.

Two honest cautions. First, delta is not an ordinary function — it is a distribution — so E is only required to satisfy L E = delta in the distributional sense, and E itself is typically singular at the source point (it blows up there). Second, a fundamental solution is not unique: you can add any solution of the homogeneous equation L v = 0 and still have L(E + v) = delta. Which one you want depends on side conditions — decay at infinity, or causality in time (the retarded versus advanced choice). The fundamental solution lives in free space with no boundaries; once a boundary appears, you correct it into a Green's function.