Step 1 - Type of battery.
The mercury cell is a PRIMARY BATTERY (a primary cell). The reaction occurs only once
and the cell CANNOT be recharged or reused after the reactants are used up.
It is a small button-type cell used in low-current devices such as wrist watches,
hearing aids, calculators and cameras.

Step 2 - Construction.
Anode : zinc-mercury amalgam, $Zn(Hg)$
Cathode : a paste of mercury(II) oxide, $HgO$, and carbon
Electrolyte : a paste of KOH and ZnO (alkaline)

Step 3 - Why it is more advantageous than a dry cell.
The mercury cell gives a CONSTANT (steady) POTENTIAL of about 1.35 V THROUGHOUT ITS
USEFUL LIFE.
The reason is that the overall cell reaction involves ONLY SOLIDS AND LIQUIDS - no ion
in the solution is consumed or produced. Since the composition of the electrolyte does
not change, the cell potential (which depends on concentration through the Nernst
equation) stays constant.
In contrast, in a Leclanche dry cell the $NH_4^+$ ion is consumed and $NH_3$ and $Mn(III)$
species accumulate, so the concentrations change continuously and the voltage FALLS
steadily from about 1.5 V as the cell is used.

Step 4 - Write the reactions.
At the anode (oxidation):
$$Zn(Hg) + 2OH^-(aq) \longrightarrow ZnO(s) + H_2O(l) + 2e^-$$
At the cathode (reduction):
$$HgO(s) + H_2O(l) + 2e^- \longrightarrow Hg(l) + 2OH^-(aq)$$

OVERALL CELL REACTION:
$$Zn(Hg) + HgO(s) \longrightarrow ZnO(s) + Hg(l)$$

Note that $OH^-$ and $H_2O$ cancel out completely, which is exactly why the electrolyte
composition - and hence the voltage - remains unchanged.