Electrons don't actually flow through water by themselves. The need to be "escorted" through the water by ions. For example, a H₂O water moecule can give up one of its H atoms but keep its electron, and become OH- (the "-" means it's negatively charged). The other H₂O molecule then becomes H₃O+. The OH- and H₃O+ can both move through the water and carry electrical charges. If you put two electrodes in water and attached them to a battery, the electrical current would be carried by the OH- and H₃O+... very slowly.

Pure water is actually not a very good conductor. It's better than many other liquids, but a trillion (a 1 with 12 zeroes after it) times worse than metals like copper. You have to add electrolytes (salts, acids, or bases) to make conductive circuits. Even with electrolytes, the conductivity is still about a million times worse than copper wires. On the other hand, instead of using water to conduct electricity, you can actually build primitive computers using the *flow* of water to make decisions, with no moving parts. This is called fluidics (like fluid electronics). Fluidics can be used in places too hostile for electronics, like the backup control system in a nuclear reactor.