Why are primitive roots called generators? I learned recently that the reason that g is commonly used to denote a primitive root is because it stands for "generator". I also know that this has something to do with the non-zero residues. However I don't understand how a g could be used to "generate" all non-zero residues.

The relevant (and cool) fact is this: if $g$ is a primitive root modulo $n$ (so that $g$ has multiplicative order $\phi(n)$ modulo $n$), then every integer in the world that is relatively prime to $n$ is congruent modulo $n$ to some power of $g$. In other words, the powers of $g$ "generate" the entire set of reduced residue classes modulo $n$. (In terms of groups: the multiplicative group of reduced residue classes modulo $n$ is cyclic, and $g$ is a generator for that cyclic group.)
