Baire's theorem is considered fundamental for functional analysis. It could be simply stated as

In complete metric spaces (e.g.) countable intersections of dense open sets are dense.

But in many textbooks, there is instead a definition of what it means for a set to be of first category (meager), of second category, or residual (comeager). Then a Baire space is defined as a space in which every residual set is dense, and Baire's theorem becomes "complete metric spaces are Baire spaces".

The terms "first category", "second category", etc. are not needed for the statement of Baire's theorem, nor do they provide any intuition for it.

What is the bigger picture for which these concepts are needed?

Wikipedia says

a meagre set (...) is a set that (...) is in a precise sense small or negligible

In what sense?

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    $\begingroup$ Although I do not claim anything about history... in all my experience there is nothing helpful or compelling or needed anywhere about "first category" or "second category" or "meagre"... That is, it's better to give the version you do, or maybe its contrapositive, in terminology that explains what's actually happening. $\endgroup$ – paul garrett Nov 12 '16 at 19:25
  • $\begingroup$ In addition to Garrett's answer, most propositions use the first statement of Baire's, e.g. Uniform Boundedness / Banach Steinhaus. $\endgroup$ – user305860 Nov 12 '16 at 19:29
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    $\begingroup$ @paul: You’re wrong about meagre: that term gets directly at the underlying intuition. The statement in terms of dense open sets is generally more useful, but that’s a different issue. $\endgroup$ – Brian M. Scott Nov 12 '16 at 19:39
  • $\begingroup$ @BrianM.Scott, I don't think "meagre" or "meager" is a contemporary colloquial U.S. English term, though. I've had many grad students over the years respond with bafflement to it. So "meagre/meager" no longer clearly suggests "small" or "thin", is part of the problem. $\endgroup$ – paul garrett Nov 12 '16 at 20:31
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    $\begingroup$ @paul: That may well be the case, but it doesn’t change the fact that the word gets directly at the intuition. Of course one has to know the word in order for that fact to be helpful, but that also is a different issue, one that is quickly resolved by giving the non-technical definition of meagre. $\endgroup$ – Brian M. Scott Nov 12 '16 at 20:40

The term meagre is the one that provides the basic intuition. A nowhere dense set is clearly small in a topologically meaningful sense: it’s so small that its closure has empty interior. A meagre set is small in the sense that it can be expressed as the union of just countably many nowhere dense sets. That’s not quite so small as being nowhere dense, since a meagre set can be dense in the whole space (e.g., $\Bbb Q$ in $\Bbb R$), but this is still clearly a notion of smallness. And it’s a nicer one to work with than the basic notion of nowhere dense set, because the meagre sets form a $\sigma$-ideal: all subsets and countable unions of meagre sets are still meagre. (The nowhere dense sets merely form an ideal: all subsets and finite unions of nowhere dense sets are nowhere dense.)

The Baire category theorem then says that if a space is nice enough (complete metric space, locally compact Hausdorff space, etc.), then it isn’t small (i.e., meagre).

More generally, any ideal or $\sigma$-ideal $\mathscr{I}$ of subsets of some set $X$ can be considered a notion of smallness for subsets of $X$. Sets in $\mathscr{I}$ are small; complements of sets in $\mathscr{I}$ are large; and all other subsets are neither small nor large. Some of these notions of smallness turn out to be very useful, and the $\sigma$-ideal of meagre sets of a topological space is one of the useful ones. The ideal of non-stationary subsets of a cardinal of uncountable cofinality is another one that turns out to be very useful.

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    $\begingroup$ The book Measure and Category by Oxtoby gives a nice analogy between meagre sets and sets of measure 0. These also form a nice $\sigma$-ideal by the axioms of measure theory and normally it is assumed as an axiom that $\mu(X) \neq 0$ for a non-trivial measure, so we Baire-ness gives the topological notion of small a sort of equal footing to the analytical notion of measure $0$. Oxtoby nicely expores in this book where analogies hold and where the analogy breaks down. This might help with intuition. $\endgroup$ – Henno Brandsma Nov 13 '16 at 12:32
  • $\begingroup$ @Henno: I can't believe that Measure and Category didn't even cross my mind when I wrote that! Yes, absolutely. (Though with my background I think that I learned more about measure from it than I did about category!) $\endgroup$ – Brian M. Scott Nov 13 '16 at 14:33

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