The quotient object $Q$ of a congruence (an internal equivalence relation) $E$ on an object $X$ in a category $C$ is the coequalizer $Q$ of the induced pair of maps $E \to X$.
If $E$ is additionally the kernel pair of the map $X \to Q$, then $Q$ is called an effective quotient (and $E$ is called an effective congruence, with the map $X \to Q$ being an effective epimorphism).
Sometimes the term is used more loosely to mean an arbitrary coequalizer. It may also refer to a co-subobject of $X$ (that is, a subobject of $X$ in the opposite category $C^\op$), without reference to any congruence on $X$. Note that in a regular category, any regular epimorphism (i.e. a “regular quotient” in the co-subobject sense) is in fact the quotient (= coequalizer) of its kernel pair (actually, we can prove this under weaker hypotheses; see below).
As we have said, there are various notions of quotient object. Let us consider the most general one, so that $Quot(X)$ of an object $X$ denotes the poset of co-subobjects of $X$, in other words the posetal reflection of the preorder of epis $X \to Q$ which is a full subcategory of the co-slice category $X \downarrow \mathbf{C}$. A regular quotient then refers to a regular epi $X \to Q$.
On the other hand, let $Rel(X)$ be the poset of relations on $X$, i.e., the poset of subobjects of $X \times X$, or in other words the posetal reflection of the preorder of monos $i = \langle e_1, e_2 \rangle: E \rightarrowtail X \times X$ which is a full subcategory of the slice category $\mathbf{C} \downarrow X \times X$.
Between $Quot(X)$ and $Rel(X)$ there is a relation $\perp$ where $q \perp \langle e_1, e_2 \rangle$ means exactly $q \circ e_1 = q \circ e_2$. If the coequalizer $coeq(e_1, e_2)$ of the parallel pair $e_1, e_2: E \rightrightarrows X$ exists, then by definition we have $coeq(e_1, e_2) \leq q$ iff $q e_1 = q e_2$. On the other hand, if the kernel pair $\ker(q)$ of $q$ exists, then by definition we have $q e_1 = q e_2$ iff $\langle e_1, e_2 \rangle \leq \ker(q)$.
This indicates in a category which admits coequalizers and kernel pairs, we have
$coeq: Rel(X) \to Quot(X)$ is left adjoint to $\ker: Quot(X) \to Rel(X)$.
Or, in other words, that $\ker$ and $coeq$ set up a Galois connection between $Quot(X)^{op}$ and $Rel(X)$.
Restricting consideration to kernel pairs only of epis, or coequalizers only of jointly monic pairs, is no real restriction in the presence of epi-mono factorizations:
In a category where every morphism $f: A \to B$ has an epi-mono factorization $f = i \circ q$, we have $\ker(f) = \ker(q)$. Similarly, for a pair $f, g: X \rightrightarrows Y$, we have $coeq(f, g) = coeq(e_1, e_2)$ where $\langle f, g \rangle: X \to Y \times Y$ factors as an epi $p: X \to E$ followed by a mono $\langle e_1, e_2 \rangle: E \to Y \times Y$.
We prove just the first statement; the second is proven similarly. It suffices to observe that the same class of jointly monic pairs $(e_1, e_2)$ are coequalized by $f$ as by $q$; the kernel pair is by definition the maximum of this class. If $q e_1 = q e_2$, then by applying $i$ to both sides we deduce $f e_1 = f e_2$. If $f e_1 = f e_2$, i.e., if $i q e_1 = i q e_2$, then $q e_1 = q e_2$ by monicity of $i$.
Suppose $\mathbf{C}$ is a category with coequalizers and kernel pairs and where every morphism has an epi-mono factorization. Then every regular epi $q$ is the coequalizer of its kernel pair: $q = coeq \circ \ker(q)$. And every kernel pair is the kernel pair of its coequalizer: $i = \ker \circ coeq(i)$.
We just prove the first statement; the second is proved similarly. We have of course a counit $coeq \circ \ker(q) \leq q$. On the other hand, if $q = coeq(f, g)$ (where we may assume $\langle f, g \rangle$ is monic by the lemma), then we have a unit $\langle f, g \rangle \leq \ker \circ coeq(f, g) = \ker(q)$; applying $coeq$ to each side, we have $q \leq coeq \circ \ker(q)$, as desired.
Constructing quotient objects in an elementary topos $\mathbf{E}$, starting from one or another standard definition (e.g., finitely complete category with power objects) that doesn’t already mention colimits, is not trivial.
The standard approach seen in textbooks (see for example Sheaves in Geometry and Logic), apparently first introduced by C.J. Mikkelsen but first published by Paré, is to prove that the contravariant power object functor $P \colon \mathbf{E}^{op} \to \mathbf{E}$ is monadic. It follows that $P$ reflects finite limits in $\mathbf{E}^{op}$ from limits in $\mathbf{E}$, but finite limits in $\mathbf{E}^{op}$ are of course finite colimits in $\mathbf{E}$.
This elegant approach does involve a fair amount of categorical machinery (a monadicity theorem, Beck-Chevalley conditions, and consideration of up to six applications of the power object functor), making it a challenge to get across intuitively in say an undergraduate course.
Other approaches that are closer to naive or common sense set-theoretic reasoning are possible. In the case of quotient objects, to form the coequalizer of a parallel pair $f, g: X \rightrightarrows Y$, we outline a possible path to take (see also at quotient type – from univalence):
Construct enough of the internal logic to make available logical operators $\wedge, \Rightarrow, \forall$.
Construct an internal intersection operator $\bigcap: P P X \to P X$ via the formula $\bigcap \Phi = \{x: X\; |\; \forall_{S: P X} \Phi \ni S \Rightarrow S \ni x\}$.
Construct the image of a map $f: X \to Y$ by taking the internal intersection of all subobjects of $Y$ through which $f$ factors.
To construct the coequalizer of $f, g: X \rightrightarrows Y$:
Notice that what these steps collectively do is form the object $Q$ of equivalence classes of the equivalence relation generated by the relation $f(x) \sim g(x)$, which is exactly what we would do in ordinary set theory. Full details will appear elsewhere.
These notions have generalizations when $C$ is an (∞,1)-category:
an equivalence relation is then a groupoid object in an (∞,1)-category
it has an “effective quotient” if it is deloopable.
For instance an action groupoid is a quotient of a group action in 2-category theory.
In type theory/homotopy type theory the analogous concept is that of quotient types.