What is Multipole Expansion?
Multipole expansion is the approximation of any electrical charge distribution \( \rho \) by polynomials. The multipole expansion is therefore a Taylor expansion of the charge distribution. It approximates a function (here: \(\rho\)) by polynomials. The more polynomials are added, the better the approximation.
Each polynomial corresponds to a specific charge arrangement:

The first polynomial represents a monopole. (as if the entire charge distribution were concentrated in a single charge point).

The second polynomial is a
dipole . 
The third is a quadrupole.

The fourth is an octupole and so on.
In this way, for example, the electric potential can be calculated using the approximated charge distribution \(\rho\), whereby the analytical solution of the problem would not be possible without this approximation.