The "Axiom of Choice" is one of those simple-seeming notions that is difficult to fully-grasp... or at least grasp its profound implications (I don't claim to). It probably requires a fairly deep grounding in set theory in order to fully wrap your brain around its significance.
Anyway, here is an older post I recently ran across that just sort of struck my fancy on the topic, so maybe it will yours as well (even though the author, Adam Bohn, admits at one point that the axiom of choice falls into the category of “things it is best to not think about”!):
Meanwhile, the best overall treatment of the Axiom of Choice (for layfolks) that I've seen on the Web is here (with plenty of further links):