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Spherical harmonics are functions on the surface of a sphere which fulfill the same role that sine function amplitudes and phases (of various fractions of a period) provide in typical Fourier series equivalents of periodic functions. A series of these individual harmonics/functions (each individual harmonic represented by a coefficient) matches some given function over the surface of the sphere, i.e., it is a series expansion of that given function, specifically, a multipole expansion. Taking a finite portion of the series can approximate that the given function. Such harmonics provide a means of summarizing what is known of some physical value that varies over surface of the sphere.
Laplace spherical harmonics (a common type, often what is meant by spherical harmonics) effectively divide the sphere into portions, portioning the sphere by number of meridian divisions and number of latitude divisions. The harmonic's mode is designated by two numbers, l and m, l (the degree, multipole moment or multipole number) being a natural number indicating the number of latitude-like divisions, and m (the order or azimuthal number) being a natural number indicating the number of meridian-like divisions, there being no more meridian divisions than latitude divisions. For example, for l=1, m=1, the sphere is divided along an equator-like line and a meridian-like line, resulting in four portions. (For example, an approximate description of some function over a sphere could be stated as coefficients for each combination of l and m such that l is 0, 1, 2, 3, or 4 and m is any number from 0 to l, i.e., a total of 15 coefficients.)
Spherical harmonics are used for describing gravitational fields of planets (e.g., by a gravitational potential model; the above-used letters l and m probably grew out of analysis Earth's gravitational field in terms of latitude-divided and meridian-divided modes). They are also used in describing seismology (including asteroseismology), and are of interest in the theory of core collapse supernovae. They are also used in characterizing the distribution of the cosmic microwave background (CMB) variations around the celestial sphere (CMB anisotropies). They can be used in characterizing weather around a world. They are also used within a technique for solving some types of differential equations.