What is the intuition behind covering spaces? I've come to study this definition and become interested on the intuition behind it mainly because of the study of spinors, motivated by Quantum Mechanics.
The definition of covering space is as follows:

Let $X$ be a topological space. A covering space of $X$ is a topological space $C$ together with a continuous surjective map $p: C\to X$ such that for each $x\in X$ there is an open neighborhood $U$ of $x$ such that $p^{-1}(U)$ is the disjoint union of open sets which are homeomorphic to $U$ through $p$.

The definition means that for each $x\in X$ there's $U\subset X$ which is open with $x\in U$ which that there are $\{V_{\alpha}\subset C : \alpha \in A\}$, with all of the $V_\alpha$ open, satisfying $V_{\alpha}\cap V_{\beta}=\emptyset$ when $\alpha\neq \beta$ and $p | V_\alpha$ being one homeomorphism between $V_\alpha$ and $U$ with
$$p^{-1}(U)=\bigcup_{\alpha \in A}V_\alpha.$$
Now, although the definition is fine, I want to get some intuition about it.
When we define covering spaces, what is the intuitive thing which we are really turning rigorous with a precise definition? What is really the idea behind this definition?
I really couldn't get a nice intuition regarding this. Also, what is the importance of this definition, in the sense of when do we expect to see this concept becoming useful?
 A: Once you have learned about the fundamental group of $X$ (denoted $\pi_1(X)$), then you will learn many useful algebraic topology theorems relating subgroups of $\pi_1(X)$ to covering spaces of $X$. For example, for each integer $k \ge 1$ there is a correspondence between subgroups of $\pi_1(X)$ of index $k$ and covering spaces of $X$ for which the degree of the covering map equals $k$. The theorem which describes this correspondence precisely is sometimes called the "fundamental theorem of covering spaces". I would expect any textbook that contains the definition of covering spaces to discuss these matters in detail, such as Munkres' "Topology" or Hatcher's "Algebraic Topology".
As an example theorem related to your interest in spinors, if $n>3$ then the fundamental group of $SO(n,\mathbb{R})$ is cyclic of order 2. In any cyclic group of order 2, the trivial subgroup has index 2. Combining that simple fact with the theorem mentioned above, $SO(n,\mathbb{R})$ has a degree 2 covering space that corresponds to the trivial subgroup of the fundamental group, and this double covering space itself has trivial fundamental group. This space, by definition, is $Spin(n)$. 
If you want to garner some intuition for the definition itself, I would suggest careful study of the path lifting lemma and the homotopy lifting lemma. These two lemmas are key tools for application of covering spaces, and are used in proving the fundamental theorem of covering spaces. Understanding the proofs and the application of these lemmas might help you to develop the intuition you ask for.
