The actual charges on the ions are what attracts them to the electrodes (that are charged electrically) the positive ions migrate to the negative elctrode and vice versa.
When a negative ion arrives at the positive electrode it has its extra electron stripped off by the electrode. This electron continues on its journey to the the battery just like normal electricity (all electrons are, of course, indistinguishable)
When a positive ion arrives at the negative electrode it finds billions of electrons just waiting to fill up its positive 'hole'. One of these extra electrons jums onto the positive ion cancelling it out.
The overall effect of these two processes, or electrode reactions, is that one electron has left the catrhode and one electron has arrived at the anode. From the point of view of the battery and external circuit one electron has left the negative side and has arrived at the positive side. i.e. a current has flowed around the circuit.
Nothing actually flows across the electrolysis cell, the electrons are absorbed by one electrode and produced at the other by two different reactions. For these reactions to be able to occur the ions in the cell must be able to move, i.e. in solution or molten.