Definition

A generator $g$ of a Finite Field $\mathbb{F}_{q}$ is an element of order $q-1$; equivalently the powers of $g$ run through all the elements of $\mathbb{F}^{*}_{q}$.

Intuition

So this generator concept proves that our finite field is of a cyclic nature. It also, in laymans terms, is a number that when taking powers up until its order, generates the distinct values that construct the set $\mathbb{F}_{q}$ again. So we “generate” $\mathbb{F}_{q}$.