Il pleut des cordes is the exact equivalent to "it's raining cats and dogs" in terms of imagery and frequency and a great classic. These sort of phrases never have exact equivalents because they are rooted in the History
and traditions of a country.
Il pleut des cordes is by far the most frequent colloquial expression to say the it's raining heavily. It conveys the optical illusion of the raindrops being all attached we seem to see one continuous flood coming down from the sky.
Il pleut comme vache qui pisse* is probably a more "vivid image" but not so frequent, and a different register, it's a lot more informal.
One vivid image is probably il pleut des hallebardes. It is older than il pleut des cordes and does not sound informal, but it is much less widely used. Victor Hugo used it in Les Misérables :
il lansquine, il pleut, vieille figure frappante, qui porte en quelque sorte sa date avec elle, qui assimile les longues lignes obliques de la pluie aux piques épaisses et penchées des lansquenets, et qui fait tenir dans un seul mot la métonymie populaire : il pleut des hallebardes.
Other images are il pleut des seaux (which I gather from Plus jamais quoi encore's answer seems to be more frequent in Quebec French), as is il pleut des clous.
* It's usually comme vache qui pisse, no indefinite article.