This is probably too advanced, but there is Roland Fraïssé's book (which I'm surprised no one has mentioned yet):
Roland Fraïssé, Theory of Relations, Studies in Logic and The Foundations of Mathematics #118 [Revised Edition is #145], North Holland, 1986, xii + 397 pages [Revised Edition is 2000, ii + 451 pages].
Review of 1986 edition by Arnold William Miller, Bulletin of the American Mathematical Society (N.S.) 23 #1 (July 1990), pp. 206-209.
Review of 2000 edition by Peter van Emde Boas, Nieuw Archief voor Wiskunde (5) 5 #3 (September 2004), p. 251. [Boas's review is in English.]
(added the next day)
More useful, I think, would be to gather up a lot of undergraduate level set theory texts (Enderton, Schaum's outline, Dalen/Doets/De Swart, Devlin, Hrbacek/Jech, Monk, Roitman, Vaught, etc.) and compile a list of basic results about relations from the text material and the exercises (most will probably be in the exercises). I've often used this method to learn something new. In the U.S. you can find many such books in most any college library under the Library of Congress headings QA 9 and QA 248. As you compile and organize the results, you'll become better acquainted with subject, and sometimes you'll even come up with some new results on your own by extending ideas in the results you have. (In my case, I almost always later come across my "original result" published somewhere, usually as an exercise in a book or as an aside in a research paper.)
(ADDED 10 YEARS 3 MONTHS AFTER ORIGINAL ANSWER)
I recently came across a book that provides a nice contrast to Fraïssé's very advanced book. Indeed, probably anyone with the background for Fraïssé's would already know about it. The book below is suitable for undergraduates, even strong high school students, and is freely available on the internet at archive.org. I suspect most people reaching this web page from an internet search would find the book below much more helpful than Fraïssé's book.
Yuli (Julius) Anatolʹevich Schreider, Equality, Resemblance, and Order, translation of 1971 Russian edition by Martin Greendlinger, MIR Publishers, 1975, 279 pages.
Review of 1975 English edition by José Antonio Robles, Revista Hispanoamericana de Filosofía 11 #31 (April 1979), pp. 135-138 (review in Spanish).
analysis
studies functions but in a little different way. I'm aware that function is a special kind of relation. But relations are really interesting, they create various patterns - you can draw them as hasse diagrams (relations that create orderings), graphs are relations between vertices. It's interesting to study the properties of these patterns and structures. Maybe relations are only formalism for other branches of math, but maybe more care was devoted to them and this is what interests me. $\endgroup$