An example is the "trivial" quotient: Consider a complex $B_*$ and $A_*$ is a subcomplex of $B_*$. Then, at each dimension we have a short exact sequence
$$0\to A_n \to B_n \to B_n/A_n\to 0$$
which together give a sort exact sequence of chain complexes
$$0\to A_*\to B_*\to B_*/A_*\to 0.$$
Now that may sound silly (as it is simply the definition), but a more detailed example can be $B_*=S_*(X)$ the singular chain complex of a space $X$, and $A_*=S_*(Y)$ the singular chain complex of a subspace $Y$ of $X$. In this case $B_*/A_*=S_*(X,Y)$ is the relative chain complex of the pair $(X,Y)$. And we have the SES
$$0\to S_*(X)\to S_*(Y)\to S_*(X,Y)\to 0.$$
Another example is to look at $B_n$ as
$$\cdots \to \mathbb Z \stackrel0\to \mathbb Z\stackrel{\times 2}\to \mathbb Z\stackrel{0}\to 0.$$
$A_*=2B_*\cong \bigoplus \mathbb Z$. Now this settings give the short exact sequence
$$0\to\bigoplus\mathbb Z\stackrel{\times 2}\to \bigoplus \mathbb Z\to \bigoplus \mathbb Z/2\to 0.$$
Finally, it's worth mentioning that the SES of chain complexes are significant when we consider the Long exact sequences of homologies.
For example, the first example here gives the relative homology $H_*(X,Y)$ and also cellular homology theory. The second example gives the Bockstein operations.