In particular he studied Watt's mechanism, which was an integral component of steam engines powering the industrial revolution in Western Europe. Its optimal configuration wasn't really well understood at the time which led to practical problems. Chebyshev traveled from Russia (which wouldn't really enjoy an industrial revolution till much later) to Western Europe and discussed with experts and people who operated these engines. He brought back to Russia with him notes and experimental data, and those informed the development of what would later be known as minimax theory, and Chebyshev polynomials which provide polynomial solutions to minimax problems.

In the course of developing that theory he founded the modern field of approximation theory, and the St. Petersburg school of mathematics. I think his approach of using applied problems and techniques to inform the development of pure math deeply influenced the whole of Soviet and Slavic mathematics in the century that followed

(and yes, the book by Karl-Georg Steffens is beautiful!)

Edit: To answer the grandparent's question, aside from things directly invented by Chebyshev or his students, often things are called "Chebyshev" when there's either a Chebyshev polynomial or a minimax problem lurking in the background