So I came across this phrase in my abstract algebra textbook:
The integers $\mod n$ also partition $\Bbb Z$ into $n$ different equivalence classes; we will denote the set of these equivalence classes $\Bbb Z_n$. For example, for integers modulo 12, $\Bbb Z_0 = \lbrace ..., -12, 0, 12, 24, ...\rbrace$ and $\Bbb Z_2 = \lbrace ..., -10, 2, 14, 26, ...\rbrace$
I am trying to trace it back to the original $X$ set, i.e. if the definition of an equivalence class is $$[x] = \lbrace y \in X; y \sim x \rbrace$$ provided $x \in X$, what is the $X$ in this case?
Consider the set of natural numbers $\mathbb{Z}$ and a natural number $n$. Your book says that the set of all natural numbers can be divided into $n$ different equivalent classes. Indeed, all these classes are denoted as $[0]_n,[1]_n,...,[n-1]_n$.
Now to answer your question, consider some equivalent class $[3]_n$ for example. Let us remind ourselves the definition of $[3]_n$. $[3]_n=\{x\in \mathbb{Z}:x \equiv3 (\mod n)\}$. Notice that each element in set $\mathbb{Z}_n$ actually represents a class of integers.
You are also asking what exactly is $\mathbb{Z_n}$. By definition, $Z_n$ is set of congruence classes while $\mathbb{Z}$ is the set of all integers. In order words, if you want to list all the elements in $\mathbb{Z}$, there are infinitely many elements since there are infinitely many integers. However, if you want to list all the elements in $\mathbb{Z}_n$, that is doable and there are only $n$ elements.