How does a simple alkaline battery work?
An alkaline battery (or more precise alkaline manganese cell) consists of a zinc anode, a $\gamma\!-\!\ce{MnO2}$ cathode and $\ce{KOH}$ (in paste form) electrolyte. Zinc is the negative electrode and $\ce{MnO2}$ the positive electrode.

During battery discharge, zinc at the anode is oxidized and dissolves in the electrolyte as hydroxozincate. When the electrolyte is saturated with zincate, zinc oxide starts to precipitate. With continued discharge and decreasing $\ce{OH-}$ concentration, zinc hydroxide is formed, which can slowly dehydrate to zinc oxide.

\begin{aligned}\ce{
Zn + 4OH- &amp;-&gt;~ [Zn(OH)4]^{2-} + 2e- \\
[Zn(OH)4]^{2-} &amp;-&gt;~ ZnO + 2OH- + H2O\\
Zn + 2OH- &amp;-&gt;~ Zn(OH)2 + 2e- \\
Zn(OH)2 &amp;-&gt;~ ZnO + H2O\\
}\end{aligned}

At the cathode, $\ce{MnO2}$ is in turn reduced, first to $\ce{MnO(OH)}$. When the battery is discharged slowly, this can be further reduced to manganese(II) hydroxide.

\begin{aligned}\ce{
MnO2 + H2O + e- &amp;-&gt;~ MnO(OH) + OH- \\
MnO(OH) + H2O + e- &amp;-&gt;~ Mn(OH)2 + OH- \\
}\end{aligned}

The overall reaction, when only considering the first reduction step at the cathode, is:

$$\ce{Zn + 2MnO2 + 2H2O + 2OH- -&gt;~ [Zn(OH)4]^{2-} + 2MnO(OH)}$$

During discharge, zinc is irreversibly oxidized and water is consumed. As a consequence, the battery is "dead" when the electrolyte is dried out and/or all zinc has dissolved.

Sources: 1, 2