Example. Note first that for any module $A$, there are unique module homomorphisms $0\to A$ and $A \to 0$. If $A$ and $B$ are any modules then the sequences $0\to A \xrightarrow{\iota} A \oplus B \xrightarrow{\pi} B \to 0$ and $0\to B \xrightarrow{\iota} A \oplus B \xrightarrow{\pi} A \to 0$ are exact, where the $\iota$‘s and $\pi$’s are the canonical injections and projections respectively. Similarly, if $C$ is a submodule of $D$, then the sequence $0\to C \xrightarrow{i}D \xrightarrow{p} D/C \to 0$ is exact, where $i$ is the inclusion map and $p$ the canonical epimorphism. If $f: A\to B$ is a module homomorphism, then $A/\text{Ker} f$ [resp. $B/\text{Im} f$] is called the coimage of $f$ [resp. cokernel of $f$] and denoted $\text{Coim} f$ [resp. $\text{Coker} f$]. Each of the following sequences is exact: $0\to \text{Ker}f \to A\to \text{Coim} f\to 0$, $0\to \text{Im}f\to B\to \text{Coker}f\to 0$ and $0\to \text{Ker}f\to A \xrightarrow{f} B \to \text{Coker} f\to 0$, where the unlabeled maps are the obvious inclusions and projections.
Question: Why we put zero module $0$ in left and right side of exact sequence of module homomorphism? What exactly zero module $0$ is? It could be $\{ 0_R \}$, $\{ 0_A \}$, $\{0_D+C\}$, etc.
In category theory objects are usually considered up to isomorphism. The zero module is the zero object in the category of $R$-modules (i.e. for any $R$-module $M$ there exist unique morphisms from $M$ to $0$ and from $0$ to $M$).
It is very easy to show that two zero objects are necessarily isomorphic, so regarding your question the zero module is any zero object in the category of $R$-modules. Note in fact that whatever "zero module you choose", the morphisms $0\to M$ and $M\to 0$ are uniquely determined.
If you think about it, working in the cateogry of $R$-modules, when you have an exact sequence $0\to A\xrightarrow{f} B\xrightarrow{g}C\to 0$, this is just a way of saying that $f$ is injective and $g$ is surjective.