But the moment we use the hydrogen and oxygen we've extracted from the water, by burning it to make a car run or whatever, then you get the water back that you started with.
That's how it works - the premise of hydrogen power is simply using the energy released when hydrogen burns in oxygen to produce water. It just so happens that hydrogen + oxygen is a simple and very exothermic (releasing energy as heat) reaction. Conventional fossil fuels like heptane (gasoline at its most basic) release a lot more energy in theory, but they need a really pure oxygen supply to burn efficiently, and even then they sometimes undergo incomplete combustion, producing carbon monoxide and carbon particles (soot). That's not to mention the other by-products produced by hydrocarbon-burning engines. Nitrogen in the air starts reacting with oxygen in the high temperatures of the engine (I think - don't quote me because I'm probably wrong) to produce nitrogen dioxide and crap like that.
Of course, water vapour is one of the worst greenhouse gases (10x more than CO
2, iirc), but people wanting to switch to hydrogen power like to leave that little point out.
Damn, I've really gone on again. Don't take any notice of my babbling - I got a B in Chemistry this year (2% off an A

).