Very quickly:
Increase in concentration of electrolyte = increased rate of electrolysis.
The nature of the concentration... either the ions or the water will be oxidised. To find out, you need to look at the table of standard potentials. REMEMBER that as Cl- and H2O are close together, when a concentrated solution of, say, NaCl, is the electrolyte, Cl- will oxidise in preference to water, whereas in a dilute solution, water will be oxidised.
Electrodes may take part in the reaction and decompose if they are more likely to oxidise than water. Inert electrodes ensure that electrolysis of water occurs.