You have many questions, I'll try to adress them all.
A binary relation, as you read is just some set $R$ which is a subset of the cartesian product of two sets $A$ and $B$, that is $R \subseteq
A\times B$.
An example may ilustrate this:
Let $A=\{\dots,-4,-2,0,2,4,\dots \}$ (the set of even numbers), $B=\{1,3,5\}$.
Then a relation $R_1$ could be $R_1=\{(-4,1),(-4,3)(0,5)\}$
We usually denote a pair $(a,b)$ of a relation with the notation $aRb$ meaning a is related with b.
A function is a relation between two elements of two given sets condition that for each element in the domain there's one and only one image(*).
(*)That is: if $R$ is a function, $x_1\in Dom(f)$ and $y_1,y_2\in Im(f)$,
$$
x_1Ry_1 \wedge x_1Ry_2 \iff y_1=y_2
$$
By definition, any binary relation on a set $A$ is a subset of $A\times A$. That actually goes the other way around, so every subset of $A\times A$ is a relation.
In your example, if $A=\mathbb Z$, for example:
"a)$X = Z$ the relation $\equiv _n$ defined as $a \equiv _n b \iff n | b-a$"
this relation, $\equiv_n$, is actually the set $$\equiv_n = \{(a,b)\in\mathbb Z^2| n|b-a\}$$
However, because we very often talk about relations, we decided on an alternative way of writing them. If $R\subseteq A\times A$ is a relation, then instead of writing $(a,b)\in R$, we write $aRb$ to mean the same thing. So, instead of writing $$(a,b)\in \equiv_n$$
we write $$a\equiv_nb$$
which is easier to read.
So, to answer your confusion
I am still struggling to see why the equivalence relation would be the 'some set of ordered pairs of elements' as opposed to the relation between them
the relation between them is exactly the same thing as some set of ordered pairs of elements.
Best Answer
Short answer: yes (in both cases).
For instance, if you are working on $\mathbb N$, then the binary relation “$=$” is the set$$\{(n,n)\,|\,n\in\mathbb{N}\},$$whereas the binary relation “$\leqslant$” is the set$$\{(m,n)\,|\,m,n\in\mathbb{N}\wedge m\leqslant n\}.$$
If $A$ is a set, a binary relation on $A$ is subset $R$ of $A^2$. It's, in fact, the set of all pairs of elements $(a,b)$ of $A^2$ for which the relation holds. (This is not a definition, of course. I am just trying to convey the idea.)