How does the redox of a battery happen?
The zinc-carbon cell uses a zinc anode and a manganese dioxide cathode. It is manganese dioxide that takes part in the reaction, not carbon. Carbon is added to the cathode to increase conductivity and retain moisture(catalyst). It also contains a layer of ammonium chloride. Following is the overall reaction.
$$ \ce{Zn(s) + 2 MnO2(s) + 2 NH4Cl(aq) → Mn2O3(s) + ZnCl2 (aq) + 2NH3(aq)} $$
This is why the half reactions are as follows,   

Anode: $\ce{2Zn -&gt; Zn^{2+} + 2e–}$

Cathode: $\ce{2NH4+ + 2MnO2 + 2e– -&gt; Mn2O3 + H2O + 2NH3}$   

An electrolyte is a substance that produces an electrically conducting solution when dissolved in a polar solvent, such as water. Here, ammonium chloride solution acts as the electrolyte immobilized in the form of a paste. Basically its a chemical medium for flow of charge.