TITLE: What are the Axiom of Choice and Axiom of Determinacy?
QUESTION [114 upvotes]: Would someone please explain:

What does the Axiom of Choice mean, intuitively?
What does the Axiom of Determinancy mean, intuitively, and how does it contradict the Axiom of Choice?

as simple words as possible?
From what I've gathered from the Wikipedia page, my understanding is that the Axiom of Choice pretty much lets you make a completely random decision, i.e. equal probability for everything. I don't, however, understand what they're talking about regarding the sets of sets (is it more than what I just described?), and I further don't understand the Axiom of Determinancy.
Info on those would be appreciated. :)

REPLY [14 votes]: I'm late to the party but I just read the following explanation of the Axiom of Choice and thought it was good, so I'm adding it. (Quoted from Just / Weese, p. 118):

(A9)Axiom of Choice: For every family $x$ of non-empty pairwise disjoint sets there exists a set $z$ such that $|z \cap y|=1$ for each $y \in x$.
The Axion of Choice generalises the familiar process of choosing representatives. If $x$ is the family of congressional districts, then the Congress of the United States is an example of a set $z$as in (A9). Note that congressional districts are non-empty and disjoint.
What makes the Axiom of Choice so special that we even mention it specifically in the name of our theory ZFC? Its non-constructive character. The axiom assures us that there will be a Congress after the next Election Day but it does not specify who will be elected. Otherwise, there would be no point in holding elections in the first place. The latter option has always been preferred by dictators who evidently do not have much confidence in the Axiom of Choice and prefer legislatures composed of loyal supporters. 
Now suppose for the sake of argument that you seized dictatorial power of the United Sets of the Universe. You need to stay on good terms with your archrival, the Former Solid Intersection who keeps meddling in your internal affairs with evil propaganda about human rights, democracy and similar nuisances. In order to keep up appearances you need some sort of congress. So you start looking at the list of congressional districts and make your choices. But there is a problem: The United Sets of the Universe have infinitely many congressional districts. While you are busy picking your best congress, your enemies may be plotting your overturn. You better get quickly done with this congress-choosing business. In other words, you must make infinitely many choices in a finite amount of time without having the option of describing all these choices by a single formula. The Axiom of Choice says that you can somehow do this, but it does not give you much control over the outcome of the selection process.