I am having a hard time understanding the Axiom of Choice(AC).
Say I have an index set $A$ , and a collection of indexed sets {${V_\alpha}$}, where $\alpha$ is a member of $A$.
Then, does the difficulty of defining a "choice function" come from the fact that
(a) elements of each set $V_\alpha$ in the collection might not be "ordered" (in some sense) so that whoever is constructing a choice function is not sure which element to choose from each set?
OR
(b) the collection might be uncountably infinite so that there seems to be no systematic way to go through each set (without missing one) in the collection?
Or
(c) Is it both?
As an illustration of my confusion with AC, please consider the following examples.
(1) If I want to prove that the square of a real number is always non-negative, then I would begin my proof by saying that "Pick any real number $x$." Here, am I using AC? I am "choosing" an element from the set of real numbers $R$, but I am not specifying "how" so I feel that I am using AC. On the other hand, however, since I only have one set $R$, it seems intuitively obvious that I can just "grab" any element from the set without a problem.
(2) Here is the proof by Rudin of the theorem that monotonic functions have at most countable discontinuities.
When Rudin writes that "with every point x of E, we associate a rational number $r(x)$", is he using AC here? If I can somehow associate a natural number n(x) with every point x of E, would I still be using AC?
I am sorry if my questions are a bit all over the place, but I would appreciate it very much if you could help me understand AC!