Within mathematics Descartes did a great deal, but his name is most often used as the adjective in "cartesian coordinates". That refers to the way we graph things with one set of numbers labeled along one (usually horizontal) line and another along a line at right angles to the first (so they can also be called rectilinear coordinates, but cartesian seems to be used far more often) and associate pairs of numbers with points in the plane that contains those two lines. (When I was younger, cartesian was usually written with an upper-case C, but such formality seems to have disappeared now.)
By using this scheme Descartes made it possible to connect geometry and algebra or arithmetic. The power of this connection in solving problems in both fields has been enormous. He did a lot of other things that come up in a History of Mathematics class, but this is where a typical student hears his name.