Brine solution is the concentrated solution of $\mathrm{NaCl}$ in water.

 $\mathrm{NaCl} (\mathrm{aq}.)\to {\mathrm{Na}}_{\text{(aq) }}^{+}+{\mathrm{Cl}}_{\text{(aq) }}^{+}$

The cathode will attract the sodium ions from the sodium chloride and hydrogen ions from the water. The hydrogen ions gain 2 electrons and are reduced to form hydrogen molecules.

$2{\mathrm{H}}^{+} \left(\mathrm{aq}\right) + 2{\mathrm{e}}^{-} \to {\mathrm{H}}_2 \left(\mathrm{g}\right)$

$2{\mathrm{H}}_2\mathrm{O} + 2{\mathrm{e}}^{-} \to {\mathrm{H}}_2 \left(\mathrm{g}\right) + 2{\mathrm{OH}}^{-} \left(\mathrm{aq}\right)$

The anode will attract the hydroxide ions from the water and chlorine ions from sodium chloride. The chlorine ions lose 2 electrons and get oxidised.

${\mathrm{Cl}}_2 \to 2{\mathrm{Cl}}^{-} + 2{\mathrm{e}}^{-}$

${\mathrm{Na}}^{+} + {\mathrm{OH}}^{-} \to \mathrm{NaOH}$