I've written some lecture notes for a college math course "Mathematics of the Rubik's cube:. An extended version has been published by Johns Hopkins University Press: Adventures in group theory, look for the 2nd edition. (All royalties go directly to charity.) The errata for the 1st edition is here.
The older ascii version (below) can also be downloaded. They assume a high school mathematics background, some mathematical ability, and that you have a Rubik's cube. For more (online) background on group theory, see for example Prof Gaglione's Introduction to group theory or the book Applied abstract algebra.
Mathematics of the Rubik's cube: The tarball of the latex files and figures for notes which uses group theory to describe mathematical models for many mechanical permutation puzzles such as the Rubik's cube. It is an early version of the "Adventures in group theory" book mentioned above. I thank Christophe Troestler for preparing this version. For instructions, untar it and read the file cube_readme.txt.