Relation: Connection, tie between things; something that conditions several things.
This is a fundamental abstract idea, but everyone knows instances of it, like loves, hates, is greater than, is the same as, knows, believes, is an element of and many, many more, of many kinds, that may relate all kinds of things one can think of in any kind of connection one can think of to any kind of things one can think of.
1. Relations in logic
2. Relations analysed settheoretically
3. Relations as structures
It is not easy to analyse relations, and one of the odd things about human beings is that it took over 2000 years for logicians to analyse relations explicitly in logic. The merit for first seeing that there is a need for a logic of relations belongs to the 19th Century English mathematician Augustus de Morgan, who noted that up to then logic had no place for intuitively perfectly valid arguments like "if this is the head of a horse, this is the head of an animal".
After some initial work of his, the logic of relations was rather quickly set up, independently, by Frege and Peirce, who also independently (Peirce together with a pupil of his, Mitchell) developed the logic of quantifiers for it.
1. Relations in logic
In logic, a relation is expressed by a predicate with two or more subjects and may be written with "R" for the relation, and "x" and "y" for the subjects, as "R(x,y)", "xRy", "Rxy" etc. The notation is a matter of choice, and the three given ones are common.
Now this gives, within formal logic, the means for quite sophisticated analyses and terminology of relations, since one can front statements about relations with quantifiers ("for every x there is some y such that x loves y" etc.), and introduce new terminology for properties of such expressions. One example is that a relation R is symmetric iff (x)(y)(Rxy iff Ryx); another is that a relation R is reflexive iff (x)(Rxx), and quite few more.
There are many more of these properties of relations, but one thing to note is that one can and does introduce these in fact as properties of the linguistic expressions one uses for the relations, which one defines in terms of what subjects make statements that involve the relation true.
2. Relations analysed settheoretically
A further step is to analyze relations settheoretically, by taking "R" as referring to a set of pairs: {(a,b): (a,b)eR} = {(x,y): R(x,y)} whence (a,b)eR iff R(a,b). Here
"{(a,b): (a,b)eR}" = "the set of pairs (a,b) such that (a,b) is an element of R" and " "{(x,y): R(x,y)}" = "the set of pairs (x,y) such that x has R to y".
Now, one can use "R" or "{R}" as a name for either of last sets. This will be a set of pairs. Thus, if one speaks of love, the corresponding sets will be the set of pairs in which the first element is a lover and the second a beloved.
There are at least two problems with this otherwise neat analysis, that was first thought of by Frege and Peirce, it seems, who also saw the problems that follow, if not in the same terms.
One: In some imaginable simple Universe of Discourse, such as {Adam, Eve} with Adam and Eve being the only humans who happen to love and tease each other, whence on the given analysis, supposing there being only Adam and Eve, the relations of teasing and loving have just the same elements (namely (Adam, Eve) and (Eve,Adam)).
And the problem is that  in this simple universe  loving and teasing are true of just the same pairs of elements; loving and teasing are construed as sets; sets are identical if they are true of just the same elements; and so  in this simple universe  loving=teasing, which is not intuitively correct.
Two: It seems to follow that relations "are" sets of pairs, which may not be quite what one intuitively expects that a relation "is". Loving someone, one may say with some intuitive justification, just does not reduce to being a member of a set of pairs in which one is a first member and one's beloved a second member  there is, at least, more to it than that, notably the relation of being in love.
Let's first dispose of the first problem, by noting it also holds for oneplace predicates i.e. properties, and noting that for any predicate that is coextensive with another predicate in a certain domain, although intuitively the predicates have different meanings, one can at least alleviate the problem by adding individuals to the domains or adding predicates to the predicates that do distinguish them, or both.
This does not dissolve the underlying problem of the interpretation of predicates in settheory: They are identified with the set of things the predicate is true of, and not with a new sort of entity, that is unlike the things the predicate is true of, such as a really existing property or relation. But supposing there are only things and sets of things, this identification or confusion makes sense, especially since it leads to lots of formulas and definitions that also make sense, and preserve the great majority of one's intuitions about relations.
Next, we consider relations as sets of pairs (or tuples). As just mentioned, one reason for this reductive analysis of a relation to a set of the pairs (or tuples) of things that the relation is true of is that it requires no more entities than are already supposed in set theory: things (of any kind) and sets of things, that may also be sequences i.e. ordered sets.
And part of the problem is that even if one insists that a relation or property intuitively just is not the same kind of thing as the things it truly attributes, then it remains difficult to say what kind of thing the meaning of a relationterm or propertyterm would be.
3. Relations as structures
The most plausible answer seems to run like this: Relations and properties are names of structures  kinds of things that remain the same while one or more of their parts are changed, somewhat like a room, which remains the same while furniture in it is removed or replaced, or like a medicinestrip, with say 10 containers for pills that may be pushed out of them, and be put back, or replaced by other items that fit, or indeed again like one's body that remains much the same while some of its parts are renewed.
One reason this answer is plausible is that it does give an interpretation to relation terms and property terms that gives them their own kind of referent; another reason is that structures are analysed much like predicates  indeed, if P(a,b) is a statement with a twoplace predicate P true of the pair (a,b), then the structure involved may be written as: structure(P,a,b).
This in turn may be analysed using Cartesian Products: A structure is an entity that is like a Cartesian Product of which the first member is a constant, that often represents a complicated entity into which the things that satisfy it may enter as parts or a type of action in which some of the states of one thing condition some of the states of another thing.
