To rain cats and dogs. In northern mythology the cat is supposed to have great influence on the weather, and English sailors still say, “The cat has a gale of wind in her tail,” when she is unusually frisky. Witches that rode upon the storms were said to assume the form of cats; and the stormy north-west wind is called the cat's-nose in the Harz even at the present day.
The dog is a signal of wind, like the wolf, both which animals were attendants of Odin, the storm-god. In old German pictures the wind is figured as the “head of a dog or wolf,” from which blasts issue.
The cat therefore symbolises the down-pouring rain, and the dog the strong gusts of wind which accompany a rainstorm; and a “rain of cats and dogs” is a heavy rain with wind. (See Cat and Dog.) The French catadoupe or catadupe means a waterfall.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894
More on Rain from Fact Monster: