At the end, a Galois category is equivalent to the category $\pi$-sets, of finite sets with a continuous action of a profinite group $\pi$, so once you know this is easy to study Galois categories. The interesting part is that sometimes you a have a category, you can prove that it is Galois, so you have your $\pi$, but this is the only way you have to define the group. This is the method used by Grothendieck to define the fundamental group of a scheme (w.r.t. to a geometric point. used to define the fibre functor). See here link text for the details.