|Style over substance|
“”The government has this car, and it runs on WATER man
In one of the variants of the idea, some magical substance is added to water, making it able to burn, and thus turning it into a fuel that can be fed to an internal combustion engine. In another, the "fuel" is often referred to as HHO Water. In the purported generators, water is apparently changed into "HHO gas", from which electrons can be used to make cars run. Sometimes it's claimed that hydrogen gas is produced - thus making use of existing hydrogen fuel cells.
One particular example is the Genepax car, revealed in Japan in 2008. While quite tight-lipped about the specific processes, Genepax hinted that the car worked along similar lines to metal hydride reactions that produce hydrogen. Once this hydrogen was produced, it could be consumed in the same manner as working hydrogen fuel cells. However, with a metal hydride reactant being consumed in the process, the "fuel" component certainly wouldn't be water. Depending on the specific chemical reaction used, it may have been less efficient than other methods of hydrogen generation, and less environmentally friendly than even fossil fuel reactions. Genepax ceased trading and production in 2009, probably in case they were called out on their scam. The website, however, is still available.
 Stanley Meyer
The water fuel cell claimed by Stanley Meyer appeared in the 1990s and involved modifying existing combustion engines to take fuel directly. Meyer's supposed mechanism involved utilising hydrogen-oxygen reactions to power the engine and consumed electricity (in his "fuel cell") to split water into these components - overlapping it with the concept of the perpetual motion machine, as mentioned by Phillip Ball writing in the journal Nature. Meyer was convicted of fraud in 1996 over the claims made in his patents and died of an aneurysm in 1998. True Believers exploited his death as evidence of a free energy suppression conspiracy theory involving oil companies. PESWiki, for example, doesn't even refer to it as a conspiracy theory, but just says his death is "mysterious" and due to poisoning.
The apparently circular and perpetual motion issue with Meyer's technology is avoided with a handwave known as "frequency enhanced electrolysis". This is essentially claiming to re-write basic laws of science and chemistry, such as Faraday's Law of electrolysis and conservation of energy, by merely claiming that boosts in efficiency will solve the problem. Now, it is true that efficiency of physical processes can improve with resonant frequencies. Gravity, for example, exhibits resonance effects when the orbital periods of planets begin to coincide, nuclear magnetic resonance also uses resonant frequencies to tilt the magnetic moments of atomic nuclei despite the contribution of the very high power applied magnetic field that would resist such a tilt. Resonant processes are very real physical phenomena, but they do not violate conservation of energy in the same way that any water powered system must do. Resonance is only a method of increasing efficiency of a process from close to 0% to close to 100% - and any perpetual motion machine based around using water must go well above 100% to be useful.
Water is fully oxidised hydrogen in a low energy state. Water is an extremely thermodynamically stable molecule, hence its abundance on Earth and the fact that it is the product of many oxidation reactions, including the burning of hydrocarbon fuels. As a result of this, energy must be put into the water molecule to break it up and produce a hydrogen-oxygen mix, just over 280 kJ/mol. To put this value in perspective, converting a litre of water into hydrogen and oxygen would require approximately the same amount of energy as the recommended daily calorie intake (~2500 kCal) of an adult human. Conversely, burning enough hydrogen and oxygen to form a litre of water would release this much energy.
It is true that hydrogen gas will react with oxygen to produce energy and the only product of this reaction is water. This is the principle of a fuel cell. So you do get this energy back, but you cannot just magically get this from water without putting energy in first. The energy that needs to be put in to generate hydrogen and oxygen, due to the laws of microscopic reversibility and thermodynamics, is exactly the same as the energy that is produced by the hydrogen and oxygen reaction.
The popularity of the water power scam may be due to the misconception that hydrogen is some form of fuel, when really it's a form of short-term energy storage. Hydrogen fuel cell cars can only be powered by hydrogen if water is first broken into a hydrogen/oxygen mixture - and this energy is the theoretical maximum that can be obtained from the fuel cell. As we don't find hydrogen lying around naturally on the Earth's surface (the presence of oxygen sees to that) we can't mine it as a fuel in the same way we do with fossil fuels. We must, ourselves, put that energy in by some means - preferably through renewable energy. In the Genepax example above, it is suspected that a metal hydride was consumed in the reaction to produce hydrogen from water, hence it would just be transforming energy from one unstable chemical (a metal hydride) to another (hydrogen gas) before burning that to produce water. This isn't particularly miraculous chemistry as many reactive substances will split water into hydrogen and oxygen (magnesium, potassium to name two). But as these metals don't exist in their pure form in nature and need to be processed first we have to put energy into forming them before they can release that energy to form hydrogen, so we can burn that to get the energy back.
An analogy can be made that this is like burning a large amount of oil in a conventional power station, to produce the electricity needed to run equipment that produces "artificial" long chain alkane molecules in a lab, then burning those and calling it a miraculous and environmentally friendly process - in other words, total bullshit.
It's certainly possible to run a car on water, by electrolyzing it and then burning the hydrogen produced; the only problem is you'll need a second, bigger engine to supply power to the electrolyzer.
 See also
- Dihydrogen monoxide (dangerous, even if it's used as a fuel)
- Hydrogen economy, a somewhat plausible idea where cars produce water
- Simple Water Fuel - You can play "spot the scam cliché" with this website.
- Gas Millage Hoaxes
- "It runs on water man..." - Memorable skit from That 70's Show.