It's simple electrolysis.  Just put an electrical current through some water and hydrogen will form at the negative electrode and oxygen will form at the positive electrode.  The electrical current provides the energy needed to break the hydrogen-oxygen bonds and also provides the electrons needed for those atoms to bond with themselves, thus forming hydrogen and oxygen gas.  This process, however, proceeds fairly slowly unless a salt is added to the water to increase its conductivity.  However, adding sodium chloride causes some chlorine gas to form with the oxygen and some sodium hydroxide to form in the solution.  

Ive done it before and geodome is right no chlorine gas will be discharged unless the solution is concentrated because if you make a concentrated solution of sodium chloride there will be more chlorine ions present in the solution than hydroxide and so therefore chlorine gas will be discharged.My main suggestions are to use good graphite electrodes (note graphite is pencil lead ,however the real electrodes are the best) and have a good consistant power source that has both a good amperage output and a good voltage output (note two nine volt batteries seemed to work well for our science teacher).also  oxygen isn't a flammable gas it is just what combines with a fuel or hydrogen to burn or explode.If it was flammable imagine what would happen when you lit a match.

How do you seperate the hydrogen and oxygen in water into their seperate flammable gases?
Only hydrogen is a flamable gas.

In industrial elecrolysis always solution of KOH (or NaOH) is used

First and the most important - Nernst equation. That's where the standard potentials and concentrations enter the scene.

Then it becomes more complicated, as there are details (overpotential on different electrodes) that can be determined only experimentally... Sorry, no links nor books.