Leather is miraculous at polishing steel. The tiny pores of the leather grab at the rough burrs of steel on the edge of the razor and force them back into alignment. By stropping the edge of the razor along leather repeatedly, you can re-align the edge to the nearly perfect state it should have been in before the shave. By using your strop to align your blade every time you shave, you keep your edge sharp for far longer, and therefore only have to sharpen it every several months rather than every few weeks.
For beginners, bridle leather does an excellent job at a reasonable price. Horace and Jasper make an excellent basic strop in Calgary, Alberta, while Ezra Arthur makes a variety of robust two-sided bridle leather strops that will last a lifetime.
A solid step up is horsehide, which pulls on the steel more and strops more effectively.
For those who only want the best, cordovan leather is the way to go. It feels incredibly silky smooth and keeps your razor sharp better than any material on earth.