the Newtonian potential
Concentrate all your mass (or charge) at a single point and ask what gravitational (or electric) potential it produces. The answer — the 1/r potential of Newton's law of gravitation in three dimensions — is the seed from which every potential is grown. It is the response of empty space to a single point source, and by adding up such responses, weighted by the actual density of sources, you can build the potential of any distribution. This building-block is the Newtonian potential, also called the fundamental solution of the Laplacian.
Precisely, the fundamental solution is the harmonic-away-from-the-origin function whose Laplacian is a unit point source (a Dirac delta) at the origin. In three dimensions it is proportional to 1/r (the Newtonian potential, going as -1/(4 pi r) with the standard sign); in two dimensions it is proportional to log r (the logarithmic potential); in general it is a power or log of the distance to the source point. Convolving this kernel with a source density f gives a particular solution of Poisson's equation Laplacian u = f: the potential of the distribution is just the superposition of point-source potentials, each source contributing its 1/r tail, summed over the whole body. This is the integral solution of Poisson's equation in free space.
The Newtonian potential is the cornerstone of potential theory and the prototype of every Green's function and fundamental solution across PDE. It explains why far from any finite body its field looks like that of a point mass at the centre (the leading 1/r term), and its careful analysis — how smooth the potential is given the smoothness of the density — is the model calculation behind elliptic regularity. The crucial honesty: it is singular at the source. The potential blows up at the point mass, and the delta source is not a function but a distribution, so the equation Laplacian u = delta is to be understood in the distributional sense, not pointwise.
The gravitational potential of a uniform solid ball, computed by integrating the 1/r kernel over the ball, comes out equal — outside the ball — to that of a single point mass at its centre. This is Newton's shell theorem, and it falls straight out of superposing point-source Newtonian potentials.
Sum the point-source potential over a density and you get the field of any body.
The kernel is singular at the source (1/r blows up, log r diverges), and the source itself is a Dirac delta — a distribution, not an ordinary function. So Laplacian (fundamental solution) = delta holds in the distributional sense; away from the source the fundamental solution is perfectly harmonic.