What is the geometrical picture of a regular cover of a topological space, $X$?
A regular cover of $X$ being a covering space $(Z,p)$ of $X$ such that, the projection of the fundamental group of $Z$ under the obvious homomorphism is normal in the fundamental group of $X$.
I know that the universal cover of $X$ is the space where there is an unfolding of all the loops in $X$. Similarly, I am trying to get a picture of what a regular cover might depict. Perhaps there is more structure to it that I might be missing. I am currently referring to Algebraic Topology - An Introduction by William S. Massey.
For reference I'll show you three different covers of a wedge of $2$ circles. The two on the left are $3$-fold covers. The first is regular because there is an isomorphism taking every vertex to every other vertex. The second is not regular. The universal cover is listed on the right. It is an infinite tree with each vertex of valence $4$. Note that it is simply connected, which is a defining feature of universal covers.