In Universal Algebra, Category Theory, Abstract Algebra and Homological Algebra, "universal properties" and "commutative diagrams" are common place. Maybe an article about them (what are, how to use them, a bit of history, good examples) could be ok.

I have put on the Homological algebra front page a (currently dead) link to a page How to use exact sequences that I intend to write, again some time in the nearish future (if no-one else gets to it first). This isn't quite what you are asking about, but is closely related. I agree that How to use commutative diagrams would be a good companion article to that one. How to use universal properties would also be a good page; it would be closely related to the ``canonicality'' page I plan to write, and also to the various tensor product pages (just because tensor products are perhaps the first construction that one encounters in algebra in which the universal property really plays a dominant role in thinking about them).

Other articles that at least touch on the notion of universality are universal constructions (where the focus is on proving that examples exist by constructing universal ones, and not on using universal properties in algebra), and group presentations. It would be very nice to be able to complement these with an article of the kind you suggest.