I guess the quintessential example, satisfying your second desiderata, is

$$ 0 \rightarrow A \stackrel{f}{\rightarrow} B \rightarrow B/f(A) \rightarrow 0. $$

For example, if $f = \mu_n: \mathbb{Z} \to \mathbb{Z}$ is multiplication by $n$, this means the following is exact

$$ 0 \rightarrow \mathbb{Z} \stackrel{\mu_n}{\rightarrow} \mathbb{Z} \rightarrow \mathbb{Z}/n\mathbb{Z} \rightarrow 0. $$

Since you asked for a big list, I'll try to restrict myself to one example per answer.

I guess the quintessential example, satisfying your second desiderata, is

$$ 0 \rightarrow A \stackrel{f}{\rightarrow} B \rightarrow B/f(A) \rightarrow 0. $$

For example, if $f = \mu_n: \mathbb{Z} \to \mathbb{Z}$ is multiplication by $n$, this means the following is exact

$$ 0 \rightarrow \mathbb{Z} \stackrel{\mu_n}{\rightarrow} \mathbb{Z} \rightarrow \mathbb{Z}/n\mathbb{Z} \rightarrow 0. $$

Another example of the same general result is that, if $C$ is finitely presented, then it fits in a short exact sequence

$$ 0 \rightarrow N \rightarrow P \rightarrow C \rightarrow 0 $$

where $N$ and $P$ are finitely generated, and $P$ is projective. Think of $P$ as the generators, and $N$ as the relations you quotient out by to get $C\cong P/N$.

Since you asked for a big list, I'll try to restrict myself to one example per answer.