Applications of Galois theory for topology

I'm not sure if this is what you're looking for, but there is a Galois correspondence for covering spaces and deck transformations that is analogous to the correspondence between intermediate fields and field extension automorphisms of a Galois extension.

The paper

R. Brown and G. Janelidze, `A new homotopy double groupoid of a map of spaces', Applied Categorical Structures 12 (2004) 63-80.

preprint here shows how Janelidze's generalised Galois theory implies the existence of a strict homotopy double groupoid of a map of spaces, generalising previous constructions; the proof is also given directly.

Edit: @sudiosus: fixed the link -thanks! I can't help too much with the question about Grothendieck and Galois theory, except to say that the book by Borceux and Janelidze on "Galois theories" does relate the extension to that from fields to rings, and the referenced paper uses an even further extension, described in the book.