Here are some texts which might be of interest for understanding the geometry of finite étale covers:
- Tamas Szamuely, Galois groups and fundamental groups.
- SGA 1 (available on arxiv).
- Michel Raynaud, Anneaux locaux Henséliens
- Deligne, P., Le groupe fondamental de la droite projective moins trois points, in: Galois groups over Q, editors Y. Ihara, K. Ribet, J.-P. Serre, MSRI Publications 16, Springer, 1989, 79-297.
The analogue of 'covering space' of a base $S$ will be finite étale morphisms $X\to S$, in the following sense: if you have a point $s\in S$, there is a 'geometric point' $\overline{s}$ (spectrum of a separably closed field) and a "tiny ball for the étale topology" which is the spectrum of a strict henselization, called the strict localization together with morphisms $\overline{s}\to S_{(\overline{s})} \to S$. Being finite étale is preserved by base change, so you'll have the base change of $X/S$ to $X_{S_{(\overline{s})}}:=X\times_S S_{(\overline{s})}$ over $S_{(\overline{s})}$ and $X_{\overline{s}}:=X\times_{S} \overline{s}$. Basic commutative algebra allows one to classify finite étale algebras over a field or a strict henselization, and you get a disjoint union of the bases in these base changes. The scheme $X_{\overline{s}}$ is the fiber of your base point, each component of which sits inside a copy of $S_{(\overline{s})}$ in $X_{S_{(\overline{s})}}$:
$\begin{matrix} X_{\overline{s}}&{\rightarrow}&X_{S_{(\overline{s})}}&{\rightarrow}&X\\ {\downarrow}& & \downarrow & & \downarrow \\ \overline{s} & \rightarrow & S_{(\overline{s})} & \rightarrow & S \end{matrix}$
so that a finite étale morphism is 'locally for the étale topology' a trivial covering.
The functor sending $X$ to $X_{\overline{s}}$ is called the fiber functor at $\overline{s}$ and the automorphism group of this functor is defined to be $\pi_1(S,\overline{s})$ - the fundamental group of $S$ at the geometric point $\overline{s}$. Each $X_{\overline{s}}$ is endowed with an action of this group, and the first step is to realize that the geometry of a cover $X/S$ is somewhat encoded by how $\pi_1(S,\overline{s})$ acts on the fibers.
Edit: Coming back to Galois theory, one can in particular show that if $K$ is a field then a finite étale $K$-algebra is just any finite product of finite separable extensions of $L$. Taking $S=Spec(K)$, a separable closure $K^{sep}/K$ gives an associated geometric point $Spec(K^{sep})\to Spec(K)$ and one finds that that $\pi_1(S,\overline{s})$ is the absolute Galois group $Gal(K^{sep}/K)$. (For example this is worked out in Szamuely's text under the name "Grothendieck's version of Galois theory" ).