As we groups and rings, we have the standard four isomorphism theorems for modules, the first of which gets by far the most use:

The First Isomorphism Theorem for Modules

Let $\varphi :M\to N$ be an $R$-module morphism. Then the map $m+\mathrm{ker}(\varphi )\mapsto \varphi (m)$ defines a module isomorphism $M/\mathrm{ker}(\varphi )\simeq \varphi (M)$.

The Second Isomorphism Theorem for Modules

Suppose $A,B$ are submodules of an $R$-module $M$. Then $(A+B)/B\simeq A/(A\cap B)$.

The Third Isomorphism Theorem for Modules

Suppose $A,B$ are submodules of an $R$-module $M$ and $A\subseteq B$. Then $(M/A)/(B/A)\simeq M/B$.

Let's see how we can prove this theorem using only universal properties. To do that, we'll show that the module $M/B$ satisfies the universal property of the module $(M/A)/(B/A)$, from which it will follow that the two are isomorphic (by a unique isomorphism!). With that in mind, let ${\pi}_{A}:M\to M/A$ and ${\pi}_{B}:M\to M/B$ denote the canonical projection morphisms (that are part of the information of the quotient). Since $\mathrm{ker}({\pi}_{B})=B$ and $A\subseteq B$, by the universal property of $M/A$ the morphism ${\pi}_{B}$ factors uniquely through ${\pi}_{A}$:

We now have our morphism ${\pi}_{A,B}:M/A\to M/B$. At the level of elements, this is simply the map that takes each coset $m+A$ to the coset $m+B$. Notice that $\mathrm{ker}({\pi}_{A,B})$ is exactly the image of $B$ in $M/A$, i.e., is $B/A$. We're already in excellent shape.

Next, suppose $P$ is a module equipped with a morphism $f:M/A\to P$ such that $B/A$ is contained in $\mathrm{ker}(f)$. Then the composition ${\pi}_{A}\circ f$ is a morphism from $M$ to $P$ such that $B\subseteq \mathrm{ker}({\pi}_{A}\circ f)$. By the universal property of ${\pi}_{B}:M\to M/B$ it follows that ${\pi}_{A}\circ f$ factors uniquely through ${\pi}_{B}$ (and hence $f$ factors uniquely through ${\pi}_{A,B}$):

We have therefore shown that every module morphism $f:M/A\to P$ with $B/A\subseteq \mathrm{ker}(f)$ factors uniquely through ${\pi}_{A,B}$, which is precisely the universal property of the quotient $(M/A)/(B/A)$ (with its canonical projection morphism).

The Fourth Isomorphism Theorem for Modules

Let $N$ be a submodule of an $R$-module $M$. Then there is an isomorphism between the lattice of submodules of $M/N$ and the lattice of submodules of $M$ that contain $N$.