Our notation is based upon Singmaster notation which is really just a specialized use of standard algebraic notation. David Singmaster denoted the basic clockwise 90-degree turns of face layers as U, D, L, R, F, and B. Multiplication can be thought of as meaning "followed by." So UR means "U times R" or "U followed by R". Exponentiation represents repetition, so RR (R followed by R, or R performed twice) can be written as "R²" (R squared). A power of -1 represents an inverse, and is normally simplified to an apostrophe or prime symbol (although that simplification is not a standard convention in mathematics).
Unfortunately, in some ways, the algebraic nature of the notation has been lost somewhat with some of the ways the original Singmaster notation has been extended over the years.