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.