Branch of math studying relations There are many branches of mathematics (analysis, algebra, group theory, logic, ...). Now, I'm interested in relations and their special kinds (like equivalence relation) and their properties. I'd like to find out more about them.
Where should I search to learn everything from the origin of the concept of relation to the knowledge about them of today? Which branch of math studies them carefully?
 A: While it is not a systematic study of relations, there is a paper by Smullyan which may be of interest to you, "Equivalence Relations and Groups ". Abstract: 

Our purpose is to show how the logic of relations can be uitilized in the study of group theory. There are some striking similarities in certain theorems in group theory and certain results about equivalence relations, and we show how the former can be derived as consequences of the latter. This transition is accomplished by means of certain ismorphism theorems, proved in considerable generality in section 2, and applied to groups in section 3. In section 1 we give several miscellaneous theorems on equivalence relations, which later turn out to have their analogues in the theory of groups.

A: The theory of binary relations over a set is in a sense the same as the theory of (non-weighted) directed graphs. The theory of symmetric relations is in a sense the same as the theory of (non-weighted) undirected graphs. Indeed, every graph can be seen as the set of vertices and the adjacency relation (which is symmetric in the case of undirected graphs).
Since relations can be composed, and the composition is associative, they are of interest in semigroup theory. If you google "semigroup of binary relations", you will find many hits. These semigroups were studied quite intensively in the 1960s.
There are connections between semigroup-theoretic properties of relations and standard notions such as transitivity or being an equivalence relation. One simple example is that every equivalence relation is idempotent with respect to composition. Indeed, every preorder is. A relation $R$ is transitive iff $R\circ R\subseteq R.$ A relation $R$ over a set $X$ is interpolative, that is $(\forall a,b\in x) ((a,b)\in R \Longrightarrow (\exists c\in X) ((a,c)\in R\wedge (c,b)\in R))$, iff $R\circ R\supseteq R.$
A: At least in the undergraduate curriculum, relations are usually studied on the way to something else. Many textbooks on combinatorics and on abstract algebra will devote a chapter to relations. Have you mastered all of the information on relations in those books? 
A: 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).
A: A book on the transition to advanced mathematics will give you a thorough (50 page) introduction to relations (as opposed to a 5 page introduction as is done in Munkres' Topology).  
I liked How to Prove It by Daniel Velleman.  It covered equivalence relations, ordering relations, closures, etc.
A: On the Calculus of Relations by Tarski
The paper contains a nice introduction to the elementary theory of binary relations (unfortunately restricted access).
A: Towards algebraic abstraction of relations, I also suggest Allegories and Relation Algebras..
