Meaning: Raining very heavily.
This is an interesting phrase in that, although there's no definitive origin, there is a likely derivation. Before we get to that, let's get some of the fanciful proposed derivations out of the way.
It has also been suggested that cats and dogs were washed from roofs during heavy weather. This is a widely repeated tale. It got a new lease of life with the e-mail message "Life in the 1500s", which began circulating on the Internet in 1999. Here's the relevant part of that:
I'll describe their houses a little. You've heard of thatch roofs, well that's all they were. Thick straw, piled high, with no wood underneath. They were the only place for the little animals to get warm. So all the pets; dogs, cats and other small animals, mice, rats, bugs, all lived in the roof. When it rained it became slippery so sometimes the animals would slip and fall off the roof. Thus the saying, "it's raining cats and dogs."
This is nonsense of course. It hardly needs debunking but, lest there be any doubt, let's do that anyway. In order to believe this tale we would have to accept that dogs lived in thatched roofs, which, of course, they didn't. Even accepting that bizarre idea, for dogs to have slipped off when it rained they would have needed to be sitting on the outside of the thatch - hardly the place an animal would head for as shelter in bad weather.
Another suggestion is that 'raining cats and dogs' comes from a version of the French word 'catadoupe', meaning waterfall. Again, no evidence. If the phrase were just 'raining cats', or even if there also existed a French word 'dogadoupe', we might be going somewhere with this one. As there isn't, let's pass this by.
The much more probable source of 'raining cats and dogs' is the prosaic fact that, in the filthy streets of 17th/18th century England, heavy rain would occasionally carry along dead animals and other debris. The animals didn't fall from the sky, but the sight of dead cats and dogs floating by in storms could well have caused the coining of this colourful phrase. Jonathan Swift described such an event in his satirical poem 'A Description of a City Shower', first published in the 1710 collection of the Tatler magazine. The poem was a denunciation of contemporary London society and its meaning has been much debated.