How does electrolysis separate hydrogen and oxygen?

Water is made up of two elements Hydrogen and Oxygen, or we can say that atoms of hydrogen and atoms of oxygen makes water.

One atom of Oxygen bonds (links) with two atoms of hydrogen to for a water molecule. The molecule can be thought of as an up side down V. To see the shape of a water molecule,

The red sphere shows Oxygen atom, and the two light blue spheres shows two hydrogen atom.

Even a small amount of water contains billions or trillions of water molecules.

Electrolysis is a process of passing electric current through water thereby splitting water molecules into hydrogen and oxygen .It is a process of breaking apart water molecules using electric current.In electrolysis of water , we use two copper wires, one twelve-volt battery, and some drinking water to do the tests. To this add a pinch of baking soda. Baking soda speeds up the breaking of water.

Place a thin piece of cardboard (a paper plate will work) over the opening of the glass. Poke two thin electrical wires, approximately 1-2 feet/ 30-60cm long, through the cardboard about 2 inches/5cm apart so that one end of the wire is approximately 2-3 inches/5-7cm submerged in the water. Wrap the other ends of the wires around the positive and negative terminals of a nine-volt battery. Bubbles will soon begin to form on the submerged ends of the wires, hydrogen gas on negative wire and oxygen on the positive wire.

At the copper wire connected to the negative terminal of battery 

H⁺ gains electron and changes to hydrogen atom, two of those Hydrogen atoms unite to form H₂ gas.

At the copper wire connected to the Positive terminal of battery 

O(2⁻) loses electrons and changes to Oxygen atom, two of those Oxygen atoms unite to form O₂ gas.