What is the story behind the Chebyshev polynomials?

The Chebyshev polynomials first appeared in his paper Théorie des mécanismes connus sous le nom de parallélogrammes (1854). The remarkable "mechanisms" described in this work can be seen in action here (click on each picture to activate it).

The context is described in MacTutor:

Chebyshev was probably the first mathematician to recognise the general concept of orthogonal polynomials. A few particular orthogonal polynomials were known before his work. Legendre and Laplace had encountered the Legendre polynomials in their work on celestial mechanics in the late eighteenth century. Laplace had found and studied the Hermite polynomials in the course of his discoveries in probability theory during the early nineteenth century. It was Chebyshev who saw the possibility of a general theory and its applications. His work arose out of the theory of least squares approximation and probability; he applied his results to interpolation, approximate quadrature and other areas.

For a more extensive account of the history of this discovery, see The theory of best approximation of functions.

An earlier, but different, approach to the questions posed and solved by next-to-be Tchebyshev Polynomials was given by Augustin-Louis Cauchy in his Cours d'Analyse (1821). See p. 230 and ff.