Comment by Sniffnoy

The axiom of choice allows you to make infinitely many arbitrary choices.

You don't need the axiom of choice to make finitely many arbitrary choices. Let's say you have a pile of indistinguishable socks in front of you. You want to pick two of them. Well -- assuming that there are at least two of them to pick -- you can pick one, and then you can pick one from what remains. If something exists, you can pick one of it, that's permitted by the laws of logic; and if you need to do that multiple times, well, obviously you can just do it multiple times. But if you need to do it infinitely many times, well, the laws of logic aren't enough to support that.

You also don't need the axiom of choice if the choices aren't arbitrary, but rather are given by some rule you can specify. There's a famous analogy used by Russell to illustrate this. Suppose you have set in front of you an infinite array of pairs of socks, and you want to pick one sock from each pair. Then you need the axiom of choice to do that. But suppose, instead, it were an infinite array of pairs of shoes. Then you don't need the axiom of choice! Because you can say, I will always pick the left one. That's a rule according to which the choice is made, so you don't need the axiom of choice. You only need the axiom of choice when the choices have some arbitrary element to them, where there isn't a rule you can specify that gets things down to just a single possibility. (Isn't the choice of left over right making an arbitrary choice? In a sense, yeah, but it's only making a single arbitrary choice!)

(The axiom that lets you do this, btw, is the axiom of separation. Or, perhaps in rare instances, the axiom of replacement, but the axiom of replacement is generally irrelevant in normal mathematics.)

So that's what the axiom of choice does. Without it, you can only make finitely many arbitrary choices, or infinitely many specified choices. If you need to make infinitely many choices, but you don't have a rule to do it by, you need axiom of choice.

[Edit: Given the article, I should note that I'm describing the role of the axiom of choice in ordinary mathematics, rather than its role in constructive mathematics. I know little about the latter.]