method of images
Stand a single charge near a flat grounded metal wall and ask what field it makes. Solving the boundary-value problem directly looks hard. The method of images is a beautiful shortcut: pretend the wall is gone and instead place a fictitious mirror-image source on the far side, chosen so that the combined free-space field of the real source and its image automatically satisfies the wall's condition. The wall is replaced by a ghost.
Concretely, to find the Green's function on a region with a simple flat or spherical boundary, you take the free-space Green's function for the real source at s and add free-space Green's functions for one or more image sources placed at reflected positions, with signs and strengths chosen to make the total satisfy the boundary condition. For the upper half-space with a Dirichlet wall at z = 0, the image of a point at (x, y, z) is a negative point at (x, y, -z); the two free-space potentials cancel on the wall, giving G(x, s) = 1/(4 pi |x - s|) - 1/(4 pi |x - s_image|). Because each image source solves the homogeneous equation inside the region (its singularity lives outside), adding it does not change the point source there but does fix the boundary.
Images give exact, closed-form Green's functions for the special geometries where reflections work: half-spaces, slabs, wedges of the right angle, the interior or exterior of a sphere, the disk. They are a staple of electrostatics (a charge above a grounded plane is attracted as if to its mirror image), heat conduction with insulated or fixed walls, and wave reflection. The method is exact when it applies, but it applies only to those symmetric boundaries; general regions need eigenfunction expansions or numerics instead.
A unit charge at height a above a grounded plane z = 0: place an image charge -1 at z = -a. The potential 1/(4 pi r1) - 1/(4 pi r2) vanishes on the plane (where r1 = r2) and solves Poisson's equation only above it. The real charge feels an attractive force, exactly as if pulled toward its mirror image.
One image charge turns a hard boundary-value problem into a two-charge free-space sum.
The method works only for boundaries with enough symmetry that a finite set of reflected sources can reproduce the boundary condition — planes, spheres, and special wedges. For a generic curved boundary no finite collection of images exists, and the trick simply does not apply.