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We figured out how to formalize computability! Oh, and by the way, you can build computers.
I'm reminded of the way lemmas proved en route sometimes turn out to be a paper's major contribution. And perhaps for related reasons - the lemma is often a practical way of solving problems that arise frequently. (Trying to think of examples... Schur's lemma is a good one.)
Zorn 's lemma is another example.
Probably a better one, from my point. Well, in the sense that IIRC it's equivalent to the separating hyperplane theorem, which I've used over and over.
Johnson-Lindenstrauss should be called the “JL miracle” rather than the “JL lemma” given how often it seems to rescue impossible-sounding ideas
What does it say? (You can tell me to look it up if you want....)
Borel-Cantelli is another good lemma.
The Yoneda Lemma. In category theory it could well be called "You need a lemma".
Szemeredi!
Szemeredi regularity lemma? One might argue that it appeared first, in a different paper, but in its first "modern" form it appeared in a paper that proved an old conjecture of Erdos and Turan from 1936. Now the Regularity Lemma represents a particular philosophy on its own.
On the same stream, another favourite of mine is the Crossing Lemma of Ajtai, Chvatal, Newborn and Szemeredi. Althought as far as I remembered it was not an auxiliary lemma in a paper. But again, it is as important as a general purpose tool than the conjecture it sought to prove.
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