spherical harmonics
Spherical harmonics are the natural 'notes' of a sphere: just as a vibrating string has a fundamental and overtones, the surface of a sphere has a discrete set of standing-wave patterns, labelled by two integers l and m. Any function on a sphere -- the angular pattern of a wavefunction, the lumpiness of Earth's gravity, the temperature map of the cosmic microwave background -- is a sum of these patterns.
The spherical harmonic Y_l^m(theta, phi) is the product of an associated Legendre function P_l^m(cos theta) with the azimuthal factor e^(i m phi): Y_l^m is proportional to P_l^m(cos theta) e^(i m phi), with l = 0, 1, 2, ... and m running over the integers from -l to +l (so 2l + 1 harmonics at each l). They are the angular solutions of Laplace's equation in spherical coordinates and the eigenfunctions of the angular part of the Laplacian: the operator L^2 acting on Y_l^m gives l(l + 1) Y_l^m, and L_z gives m Y_l^m. They form a complete orthonormal set on the sphere: the integral of Y_l^m times the conjugate of Y_l'^m' over all solid angle is delta_(ll') delta_(mm').
In quantum mechanics they ARE the shape of orbital angular momentum: l is the total angular-momentum quantum number, m its z-projection, and multiplied by a radial function they build every hydrogen-atom orbital (s, p, d, ...). Beyond the atom they are the standard basis for anything defined on a sphere -- geomagnetism, planetary gravity fields, and the multipole (a_lm) analysis of the CMB temperature that anchors modern cosmology.
The l = 0 harmonic Y_0^0 is a constant -- a featureless sphere (the s orbital). The three l = 1 harmonics are the dumbbell-shaped p orbitals, one along each axis; l = 2 gives the five d orbitals.
Rising l labels ever more finely structured patterns on the sphere.
There are two live conventions -- the Condon-Shortley phase (a factor of (-1)^m) is included by most physics texts but not all -- so signs of individual Y_l^m can differ between references. Fix one convention and stay in it.