Combustion usually has three requirements: 1)fuel, 2)oxidizer and 3)heat or source of ignition. In the case of hydrogen and oxygen, hydrogen is the fuel and oxygen is the oxidizer. Even after they are mixed, they still require a spark or other source heat to ignite the extremely rapid combustion that results in an explosion.
The process of burning in this case is the oxidation of hydrogen or combination of the hydrogen and oxygen molecules. The result of the oxidation of hydrogen, its combustion product, is simply H2O or water. The reason that water (in any state) is not flammable is that it is already the product of combustion.
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