Spherical harmonics are basically a fourier series. They're a complete orthonormal set of basis functions for functions for the unit sphere. Whereas the fourier series from calc 101 is a complete orthonormal set of basis functions on the unit interval (eg [0,1]).

In other words you can express any reasonable function on the unit sphere as a series of spherical harmonic terms. That makes them ideal for working with differential equations (eg schrodinger's equation for the hydrogen atom, or, emission from an arbitrary light source).