[Note: I see several responses posted since I started pecking this out,; instead of editing I'm just gonna post as is, even though I'm duplicating some answers.]
Hi, color photographic paper basically has three different color-sensitive layers in it. Each layer forms a different colored dye. The individual dyes are called cyan, magenta, and yellow. The layers form a dye that is roughly "opposite" to the light exposing them, like so: the layer forming cyan dye is roughly sensitive to reddish light; the magenta dye layer is sensitive to greenish light; the yellow dye layer is sensitive to bluish light. The dyes are actually formed by the action of the developer on certain precursors that already exist in each layer - in essence each layer already "knows" what color dye it can form.
It may seem strange that things happen in this seemingly backwards order, but the color FILM also behaves in the backwards manner - the net result is that the two "backward" actions combined end up producing the original color.
Here's roughly what YOU have to do: 1) buy some color photo paper, AND the special processing chemicals (the process is called "RA-4"); it uses a special color developer and a "bleach-fix," each of which you have to mix up according to the instructions. Then you have to take everything into a completely dark room. You make the very-finely controlled exposure, then process (develop, then bleach-fix, then wash) at the proper temperature for the proper time. As a note, processing unexposed paper will result in white paper. Exposing thoroughly to white light, consisting of reddish, greenish, and bluish light, will result in ALL three dyes being formed, appearing black.
In practice one fine tunes the color with small adjustments in the colored exposure. It will be very difficult to control if you don't have some specially designed equipment. As a note, the paper is pretty sensitive - if you were to accidentally turn on the lights in your darkroom for just a second or so, this is probably enough to make the paper go black. And if your exposing light doesn't have the proper color makeup then you won't be able to properly control the results.