nLab
quotient set

Given a set $S$ and an equivalence relation $\equiv$ on $S$, the quotient set of $S$ by $\equiv$ is the set $S/\equiv$ whose elements are the elements of $S$ but where two elements are now considered equal if in $S$ they were merely equivalent.

If changing the definition of equality like this is not allowed in your foundations of mathematics, then you can still define $S/\equiv$ as a subset of the power set of $S$ as follows:

(1)$$A\in S/\equiv \iff \exists (x:S),A=\{y:S\mid x\equiv y\}$$A \in S/{\equiv} \;\Leftrightarrow\; \exists (x: S),\; A = \{y: S \;\mid\; x \equiv y\}

If you don't have (extensional) power sets either, then you'll have to use setoids (in which case, perhaps you'd do better to change your terminology as described there). Alternatively, don't worry about any of this and just include the existence of quotient objects in Set as an axiom (which you have if you assume that $Set$ is a pretopos).

In any case, the element of $S/\equiv$ that comes from the element $x$ of $S$ may be denoted $[x{]}_{\equiv }$, or simply $[x]$ if $\equiv$ is understood, or simply $x$ if there will be no confusion as to which set it is an element of. This $[x]$ is called the equivalence class of $x$ with respect to $\equiv$; the term ‘class’ here is an old word for ‘set’ and refers to the definition (1) above, where $[x]$ is literally the set $A$.

Quotient sets in Set generalise to quotient objects in other categories. In particular, an exact category is a regular category in which every congruence on every object has an effective quotient object.

category: foundational axiom