The process of separating oxygen and hydrogen is known as decomposition, which means oxidation-reduction reaction. 

Oxidation means giving up electrons and reduction means gaining electrons.  Water is oxidized at the anode. The reaction is 2H2O — O2+4H+4e- where 4e- stands for 4 negatively charged electrons that are given up in the oxidation process. Water is reduced at the cathode. The reaction is 4H2O+4e- — 2H2 + 4OH- where 4e- stands for the electrons that are gained in the reduction process. In the overall decomposition reaction, the volume of H2 produced is twice the volume of O2:2H2OH — 2H2+O2.

What we have, therefore, are free negative and positive ions circulating in a water solution (the physical body is a water solution), and occasionally combining to form H2O2,otherwise known as hydrogen peroxide. So we are dealing with free hydrogen, free oxygen, and various combinations of both.