Various families of orthogonal polynomials can probably be considered fundamental in some way, but here I want to single out the Chebyshev polynomials, which seem to be the most miraculous. They have explicit formulas (T_n(x) = cos(n arccos(x)), for example), and they are closed under composition. Also, the polynomial of a given degree that has given bounds on an interval and grows as fast as possible outside the interval is a translated and scaled Chebyshev polynomial. The Chebyshev polynomials are fundamental in numerical analysis. Among their uses:

- Accurate numerical integration
- Accurate polynomial interpolation (if one is allowed to choose the evaluation points)
- Sometimes the mere existence of polynomials with the boundedness/growth properties of Chebyshev polynomials is useful, e. g., in analyzing the convergence of the method of conjugate gradients

(A little bit more on the relation to conjugate gradients: n steps of CG can be interpreted as optimizing something over the space of polynomials of degree n. The optimality of the Chebyshev polynomials in the sense of being as small as possible on [-1, 1] (stated above in a different form) is not quite equivalent to what CG requires, but it's close enough to imply that the optimal polynomial in the sense of CG must be very good (because it is at least as good as the Chebyshev polynomial).)