The Chebyshev polynomials first appeared in his paper <A HREF="http://www.math.technion.ac.il/hat/fpapers/cheb11.pdf">Théorie des mécanismes connus sous le nom de parallélogrammes</A> (1854). The remarkable "mechanisms" described in this work can be seen in action <A HREF="http://www.tcheb.ru">here</A> (click on each picture to activate it).

The context is described in <A HREF="http://www-history.mcs.st-and.ac.uk/history/Biographies/Chebyshev.html">MacTutor:</A>

> 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.