From what I know, when two electrodes are places in a electrolyte (say $\ce{H2SO4}$, which decomposes into $\ce{H}^{+}$ and $\ce{SO4}^{2-}$ in aqueous solution), the negative ion is attracted to the positive electrode and is oxidized while the cation is attracted to the negative electrode and is reduced.
This transfer of electrons from the negative electrode to the cation and from the anion to the positive electrode "completes" the circuit and electricity is conducted through the wire.
What I don't understand is how the electrolyte allows for the electrolysis of water, since the electrons are used in the redox reaction of the ions in the electrolyte in which water isn't even involved.