Pure water does not conduct electricity. But when small amount of acid say $H _{2} SO _{4}$ is added to it, water ionises. On passing electricity, it decomposes. $H _{2} SO _{4}$ being a strong electrolyte ionise completely whereas water is feebly ionised
$H _{2} SO _{4} \longrightarrow 2 H ^{+}+ SO _{4}^{2-}$
$H _{2} O \rightleftharpoons H ^{+}+ OH ^{-}$
During electrolysis, the hydrogen ions migrate towards the cathode and discharge here in the form of hydrogen gas
$2 H ^{+}+2 e^{-} \longrightarrow H _{2} \uparrow$
At anode, the concentration of $OH ^{-}$ions is too low to maintain a reaction and sulphate ions are not oxidised but remain in solution. Thus water molecules must be the species reacting at anode
$2 H _{2} O \longrightarrow O _{2}+4 H ^{+}+4 e^{-}$
The overall reactions are :
At cathode
$2 H ^{+}+2 e^{-} \longrightarrow H _{2}$
$4 H ^{+}+4 e^{-} \longrightarrow 2 H _{2}$
At anode
$2 H _{2} O \longrightarrow O _{2}+4 H ^{+}+4 e^{-}$
Overall cell reaction :
$4 H ^{+}+2 H _{2} O \rightleftharpoons 2 H _{2}+ O _{2}+4 H ^{+}$
as we see that acid used is regenerated at the end. Therefore, the whole electrolysis reaction is the dissociation of water to give oxygen at anode and hydrogen at cathode catalysed by $H _{2} SO _{4}$.