the Hermite polynomials
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The Hermite polynomials are the family that shows up glued to a Gaussian bell curve. They are the natural building blocks whenever a problem lives on the whole infinite line with a bell-shaped weight -- most famously the quantum harmonic oscillator, whose energy states are a Hermite polynomial multiplied by e^(-x^2/2).
H_n(x) is a degree-n polynomial solving Hermite's equation y'' - 2x y' + 2n y = 0. The first few (physicists' convention) are H_0 = 1, H_1 = 2x, H_2 = 4x^2 - 2, H_3 = 8x^3 - 12x. They are orthogonal on the whole real line but with the Gaussian weight e^(-x^2): the integral from -infinity to infinity of H_m(x) H_n(x) e^(-x^2) dx = 0 for m not equal to n, and equals 2^n n! sqrt(pi) for m = n. Rodrigues' formula gives them: H_n(x) = (-1)^n e^(x^2) (d/dx)^n e^(-x^2).
Multiply H_n(x) by the Gaussian e^(-x^2/2) and you get the eigenstates of the quantum harmonic oscillator, psi_n proportional to H_n(x) e^(-x^2/2), with energies E_n = (n + 1/2) hbar omega. Because almost any smooth potential looks like a parabola near its minimum, this makes the Hermite-Gaussian set the universal small-oscillation basis -- for molecular vibrations, phonons, and the modes of a quantized field. The same functions are the Hermite-Gaussian transverse modes of a laser beam.
The oscillator ground state is n = 0: H_0 = 1 times e^(-x^2/2), a pure Gaussian with energy (1/2) hbar omega. The first excited state uses H_1 = 2x, giving a wavefunction with one node -- odd, and zero at the centre.
Each oscillator level multiplies the Gaussian by the next Hermite polynomial.
Two conventions coexist: the physicists' H_n (weight e^(-x^2)) and the probabilists' He_n (weight e^(-x^2/2)), differing by a scaling. Formulas silently disagree if you mix them.