Generator

A generator is an element, or more generally a set of elements, from which the whole group can be built.

If $G = \langle g \rangle$, then $g$ is a generator of the cyclic group $G$.

More generally, a subset $S$ generates $G$ if every element of $G$ can be written as a finite word in elements of $S$ and their inverses.

One-line intuition

Generators are the building blocks from which the whole group is produced.