A multipole expansion is a type of mathematical series expansion for functions whose domain is the surface of a sphere. It is analogous to a Fourier series expansion of a periodic function, but is for a function on two dimensions over directions from a point, or equivalently, the surface of a sphere, rather than a single linear dimension.
A commonly-used type is based on spherical harmonics (harmonics over divisions of a sphere rather than over divisions of a line segment as is used for description of waves such as sound waves). The function can be real or complex, and can also be a function of the radius (as a third dimension), e.g., representing a field surrounding a specific point. The first term of the series has the name monopole, the second term is the dipole, the third, the quadrupole (or quadrapole), the fourth, octupole (or octopole: folks in different areas of application often have a preference), with further terms more often identified by number, such as 16-pole, 32-pole. The first few terms are often useful as an approximation of the function, providing a more tractable way to find useful answers. Terms such as octupole level are often used as a description of a method or calculation to assert that it produces an accuracy equivalent to using all the series terms to and including the octupole term.
Applications include description and analysis of electromagnetic fields, gravitational fields, the cosmic microwave background, and potentially any interesting property of the sky based on direction from Earth, such as any cosmic background radiation.