a potential function

Many forces in nature are lazy in a precise sense: the work they do moving you from one point to another depends only on where you start and end, not on the path. Gravity and electrostatics are like this. For such a force you can define a single scalar function — a height-like landscape — whose downhill direction is the force, and the force is conservative. That landscape is the potential. Potential theory is the study of these functions, and Laplace's and Poisson's equations are its central equations.

Precisely, a vector field F is said to have a potential u if F = grad u (or, by another sign convention, F = -grad u): the field is the gradient of the scalar u, so it points in the direction u increases (or decreases) fastest. Knowing u tells you the whole field at once. The physics then forces an equation on u: if the field is source-free (div F = 0), then div grad u = Laplacian u = 0, so u is harmonic; if there are sources of density rho (div F = rho), then Laplacian u = rho, which is Poisson's equation. That is why equilibrium potentials are harmonic away from their sources.

Potentials turn a three-component vector problem into a one-function problem, which is why they are everywhere: the gravitational potential, the electrostatic potential, the velocity potential of irrotational incompressible flow (where the velocity is grad phi and incompressibility makes phi harmonic), the magnetic scalar potential in current-free regions. The catch is honesty about when a potential exists: a field has a potential only if it is conservative, which in nice regions means curl-free; on a region with holes a curl-free field can still fail to have a single-valued potential.