fundamental solution
If a Green's function is the response of a system that has boundaries, the fundamental solution is the response of the same operator with no boundaries at all — the bare answer of the differential operator to a single point source sitting in unbounded space (or unbounded space-time). It is the purest building block: what an infinite, featureless medium does when you poke it once.
Precisely, a fundamental solution of a linear operator L (with constant coefficients, say) is any distribution E satisfying L E = delta in all of space, with no boundary conditions imposed. For the Laplacian in three dimensions, L = -nabla^2, the fundamental solution is E = 1/(4 pi r), the familiar Coulomb potential of a unit point charge; in two dimensions it is -(1/(2 pi)) ln r. For the heat operator it is the Gaussian heat kernel that spreads in time; for the wave operator it is the expanding spherical shell that encodes Huygens' principle. Because no boundaries pin it down, a fundamental solution is unique only up to adding any solution of the homogeneous equation L v = 0.
Fundamental solutions are the master keys of linear PDE theory. The full Green's function for a bounded region is built by taking the fundamental solution and correcting it — by the method of images, or by adding a homogeneous solution — so that the boundary conditions are met. They also give existence: convolving the fundamental solution with any reasonable source f produces a particular solution u = E * f of L u = f. In physics they are the propagators (heat, Schrodinger, wave) that carry an influence from a point of space-time to everywhere else.
In 3-D, -nabla^2 (1/(4 pi r)) = delta. You can check it away from the origin (where 1/r is harmonic) and confirm the delta by integrating -nabla^2(1/(4 pi r)) over a small ball and using the divergence theorem: the flux of -grad(1/(4 pi r)) through the sphere equals exactly 1.
1/(4 pi r) is the electrostatic potential of a unit point charge — physics' name for the fundamental solution of the Laplacian.
Fundamental solution and Green's function are often used loosely as synonyms, but the precise distinction is boundaries: the fundamental solution ignores them (free space), while the Green's function is the fundamental solution corrected to satisfy specific boundary conditions on a specific region.