How does electrical current move through water? Here's a little movie of it, made by Mark Petersen. A positively charged proton gets passed from one molecule to another!
This is called the 'Grotthuss mechanism' because Theodor Grotthuss proposed this theory in his paper “Theory of decomposition of liquids by electrical currents” back in 1806. It was quite revolutionary at the time, since ions were not well understood.
Something like this theory is true. But in fact all the pictures I've shown so far are oversimplified! A hydronium ion is too powerfully positive to remain a lone H₃O⁺. It usually attracts a bunch of other water molecules and creates a larger structure!
(2/n)