The axiom of choice is an axiom in set theory. It was formulated about a century ago by Ernst Zermelo, and was quite controversial at the time. It states the following:
It seems obvious: if you've got a bunch of boxes lying around with at least one item in each of them, the axiom simply states that you can choose one item out of each box. Where's the controversy?
Well, the controversy was over what it meant to choose something from these sets. As an example, let us look at some sample sets.
And therein lies the crux of the axiom. All it states is that there is some function f that can choose an element out of each set in the collection. It gives you no indication about how the function would be defined, it simply mandates its existence. Theorems whose proofs involve the axiom of choice are always nonconstructive: they postulate the existence of something without telling you how to get it.
The axiom of choice has been proven to be independent of the remaining axioms of set theory; that is, it can be neither proven nor disproven from them (unless those remaining axioms contain a contradiction, which we don't know). This is the result of work by Kurt Gödel and Paul Cohen. There are thus no contradictions if you choose not to accept the axiom of choice; however, most mathematicians accept either it, or a weakened variant of it, because it makes their jobs easier.
One of the reasons that some mathematicians do not particularly like the axiom of choice is that it implies the existence of some bizarre counterintuitive objects. An example of this is the BanachTarski Paradox which amounts to saying that it is possible to "carveup" the 3dimensional solid unit ball into finitely many pieces, and, using only rotation and translation, reassemble the pieces into two balls each with the same volume as the original. Note that the proof, like all proofs involving the axiom of choice, is an existence proof only: it does not tell you how to carve up the unit sphere to make this happen, it simply tells you that it can be done.
One of the most interesting aspects of the axiom of choice is the sheer number of places in mathematics that it shows up. There are also a remarkable number of statements that are equivalent to the axiom of choice, most important among them Zorn's lemma and the wellordering principle: every set can be wellordered. (In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the wellordering principle.)
Jerry Bona once said: "The Axiom of Choice is obviously true, the wellordering principle obviously false, and who can tell about Zorn's Lemma?". In truth, all three of these are mathematically equivalent, but the statement was amusing because it underscored the fact that most mathematicians find the axiom of choice to be intuitive, the wellordering principle to be counterintuitive, and Zorn's lemma to be too complex to form any intuitive feeling about.
Several central theorems in various branches of mathematics require the axiom of choice (or one of its weaker versions, such as the ultrafilter lemma, the axiom of countable choice[?], or the axiom of dependent choice[?]):
There are many people still doing work on the axiom of choice and its consequences. If you are interested in more, look up Paul Howard at EMU (http://www.emunix.emich.edu/~phoward/).
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