Why the set $S$ is being called a relation when the actual 'relation' is something else? Let there be two sets,
$$A=\{ \text{Jacob}, \text{Mike}, \text{Chris}, \text{John} \}$$
$$B=  \{ \text{Cindy}, \text{Amanda}, \text{Nicole}, \text{Steph} \}$$
Set $A$ has its elements as names of males.
Set $B$ has its elements as names of females.
Jacob was the husband of Cindy, Mike of Amanda, Chris of Nicole and John of Steph.
Let S be a set of ordered pairs $(a,b)$ where $a \in A$ and $b\in B$ with the relation that $a$ is the husband of $b$, i.e.
$$S=\{(a,b) | \text{  } a\in A  \text{ and } b\in B , a \text{ is the husband of b} \}$$
$$S=\{ \text{(Jacob, Cindy), (Mike, Amanda), (Chris, Nciole), (John, Steph)}\}$$
In books and other sources that I have referred the set $S$ itself is being called as a relation. Won't the relation be "a is the husband of b"?
Why $S$ is called a relation? Isn't $S$ actually a set whose elements are ordered pairs having some relation between their objects? In what context does calling S a relation makes sense?
Why is the set S being called as a relation when the actual relation is "is the husband of"?
 A: The point is to formalize the everyday notion of relation (two things having something to do with each other) and provide a precise mathematical definition. In math, once you dig the rabbit hole deep enough, things are usually defined in terms of sets. If you like, you can say that the set $S$ mathematically formalizes the notion of "is a husband of" between the people described by the sets $A$ and $B$.
But even intuitively, it should make sense as to why we define relations to be subsets of the Cartesian product (because if we want a pair to be related, then we just add that into the set).
Now, perhaps you're unhappy with the terminology because up until now you've been using the term relation in the colloquial sense, and now suddenly mathematicians have given it a formal definition. Well, that's just something one learns to live with (e.g in Probability theory an "event" has a formal definition as well: it is a set belonging to some sigma-algebra... meanwhile in Physics, an event is typically an element of the manifold which models spacetime etc).
A: Any subset of $A × B$ is a relation from $A$ to $B$. 
A relation pairs up elements of $A$ with elements in $B$.
If $a$ from $A$ is paired with $b$ from $B$ in a relation $R$, then we write $(a, b) ∈ R$ or $aRb$.
Example: $A = \{1, 2, 3\}, B = \{a, b\}$ then $R = \{(1, a), (3, b)\}$ is a possible relation.
