Motivation for Covering Spaces Once again it seems like I am introduced to a concept with very little idea why I should  care. According to Munkres, it's to find fundamental groups that are not trivial, but it's not enough to not make the rest of the section tedious.
It's not exactly a very intuitive idea. So, why should I learn about covering maps/spaces? How are they useful? 
 A: If you've thought about the logarithm in the context of complex variables, you know it must be defined with branch cuts or else be a multivalued function. If you want to "graph" the multivalued function, what you have is a Riemann surface. Indeed, Riemann surface theory is a natural outgrowth of complex analysis when you want to see the topology behind functions more clearly. Riemann surfaces are also an inherently natural concept to consider since it's simply the definition of a smooth manifold with smooth replaced by holomorphic, which we know is a stronger condition. Equivalently, it is a natural thing to consider in conformal geometry, as it is an intermediate between a smooth manifold (where we have no notion of angles or distance) and a Riemannian manifold (where we have both angles and distances), all in two real dimensions because that's what $\mathbb{C}$ is and it's the smallest dimension after the trivial $1$.
One cool way to build spaces is to start with a space, have a discrete group act on it, and then quotient by that action. The quotient map is a covering. In hyperbolic geometry we learn that every complete hyperbolic surface comes from quotienting the hyperbolic plane by a discrete group of hyperbolic isometries, which is a nice classification result.
In Lie theory, we learn that there is a functor from the category of Lie groups to the category of Lie algebras. For every Lie algebra $\frak g$ there is a unique simply connected Lie group $G$ with that Lie algebra, and all other connected Lie groups with that Lie algebra are quotients of $G$ by discrete kernels, which are precisely the images of covering homomorphisms. Given a Lie group, one may construct its universal cover as the quotient of an appropriate "path group" by the fundamental group of $G$.
Indeed, in general, one may sort-of "partition" the class of nice spaces according to their universal covers. All of the spaces covered a given simply connected one are related to each other by covering maps and form a category which is analogous to the category of algebraic extensions of a field with the algebraic closure sitting at the top, and with particular extensions having Galois groups which are analogous to fundamental groups of spaces and their deck transformations. There is an entire Galois theory of covering spaces analogous to usual Galois theory, which IIRC has use in the algebraic geometry of curves and function field theory.

It's not exactly a very intuitive idea.

My gut reaction to this is to either assume you're lying or don't understand the idea. Useful or not, the idea of a covering map should be very intuitive and beautiful which is justification enough (for me) to want to know it. Sure, buttressing obvious visual intuition with technical details and rigorous arguments involves lots of tedious bookkeeping which I don't enjoy either (in fact I tend to skim or skip them), but that's not the same as saying the idea is "not exactly very intuitive." If the definition of a covering space, and the main theorems such as the path-lifting property, are not intuitive to you then make sure you are visualizing the pictures behind them instead of just pushing symbols around on paper.
A: This is not a complete answer, but... 
A non-trivial fundamental group space means (roughly speaking) a more complicated space. Since the covering space share many properties with the original space, and because "good" spaces (with some reasonables hypothesis) always have universal covering (a covering with trivial fundamental group), to pass to the universal recovering is an important tool. After some study, you will recognize that the universal covering is a very natural space to think when you are studying a space.
