According to Allen Hatcher's Algebraic Topology the Bockstein morphism is the connecting morphism associated with a short exact sequence of abelian groups.
The two of interest are
1. $$0 \longrightarrow \mathbb{Z}_m \stackrel{m}\longrightarrow \mathbb{Z}_{m^2} \longrightarrow \mathbb{Z}_m \longrightarrow 0$$
2. $$0 \longrightarrow \mathbb{Z} \stackrel{m}\longrightarrow \mathbb{Z} \longrightarrow \mathbb{Z}_m \longrightarrow 0$$
The middle two maps aren't explained but merely left implicit. They are probably fairly elementary but I'm not sure what they are and they don't seem to be explained elsewhere in the text. What exactly are they?
The map $\mathbb{Z}_m\to\mathbb{Z}_{m^2}$ is multiplication by $m$, sending $a$ mod $m$ to $ma$ mod $m^2$. The map $\mathbb{Z}_{m^2}\to\mathbb{Z}_m$ is the quotient map, sending $a$ mod $m^2$ to $a$ mod $m$.
Similarly, the map $\mathbb{Z}\to\mathbb{Z}$ is multiplication by $m$, and the map $\mathbb{Z}\to\mathbb{Z}_m$ is the quotient sending $a$ to $a$ mod $m$.