TITLE: When can ZFC be said to have been "born"?
QUESTION [13 upvotes]: The "History" section of the Wikipedia article on ZFC isn't particularly helpful. The only thing I have understood from it is that ZFC had appeared after 1922. In what book or paper was ZFC first explicitly formulated and proposed?

REPLY [9 votes]: Since my comments on the comment section were getting a little big, I decided to expand the issue on first-order logic into an answer of its own. But first, let me preface this with a caveat, due to a paper by Gregory Moore ("Historians and Philosophers of Logic: Are They Compatible?"): it seems that establishing priority claims in cases such as this is extremely difficult, and not all too relevant. Perhaps a more interesting task is to be found in the establishment of the relation between the ideas of the many authors of the period, and how they jointly contributed to what we now know as ZFC.
Now, based on a reading of Moore's and Ferreirós's work on this, the picture seems to be the following: Zermelo initially proposed his axiom system, in a second-order guise. Later, Fraenkel proposed, rather half-heartily, the adoption of Replacement as an axiom; however, it's first real advocate with an audience was von Neumann, who also proposed Foundation. Independently, Mirimanoff also studied the well-founded portion of Zermelo's system, as well as his own version of Replacement, yet his work went largely unnoticed. At the same time, Skolem suggested the use of first-order logic as a way of making sense of Zermelo's "definite properties", and also seemed to employ a form of primitive recursive arithmetic as his meta-theory.
The issue about who proposed first-order logic as the right logic to axiomatize set theory is a bit muddled, in large part due to the point, discussed above, that there was no clear separation between first-order logic and higher-order logic at the time (the first two decades of the 20th century). In particular, it seems that the first to clearly isolate the first-order part of logic as being worthy of independent study were Weyl (largely due to his concern with predicative systems), some time during the 1910's, and Hilbert, in his lectures on logic in 1917. However, even then the importance of first-order systems was not clearly perceived, so when Skolem proposed to formalize set theory as a first-order theory, his proposal was met with skepticism. 
Interestingly, it seems that von Neumann's papers in the 1920's were a bit ambiguous on the issue. Ferreirós (Labyrinths of Thought, p. 373) writes: "His systems of the 1920's (...) seem to be intended as first-order, and certainly are formalizable within that frame. If that was his intention, von Neumann was the first mathematician to accept Skolem's (and Weyl's) views". If true, then von Neumann's 1928 paper ("Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Megenlehre", which is reprinted in the first volume of his Collected Works), which contains what is, according to Ebbinghaus, the first printed reference to the "Zermelo-Frankel" axiom system, would be a good candidate for the first first-order presentation of ZFC. In any case, it seems that, by the 1920's, this particular issue was not yet decided. It was only after Gödel proved his completeness and incompleteness theorems (so in the beginning of the 1930's) that first-order logic began to have a prominent place among other logic systems, which created a more hospitable environment for Skolem's proposal.
Indeed, in the first of a series of articles, Paul Bernays, introducing what would become known as the NBG theory, already emphasizes that adopting a first-order presentation "considerably simplifies" the resulting system. So by 1937 (the date of publication of Bernays's "A System of Axiomatic Set Theory - Part I"), Skolem's position was much more widespread. If I'd have to bet, I'd say that the first list to contain explicitly all the ZFC axioms in a first-order setting probably appeared during this time (perhaps in some textbook). Another possibility is Skolem's 1929 article, "Über einige Grundlagenfragen der Mathematik", which seems to be more detailed than earlier expositions, but which I have't had the opportunity to check out yet (my university's library has it, but I'll only go there on Tuesday).
Anyway, that's what I've been able to gather as of now. If I find anything else, I'll update this answer.