In the UK and Australia, this is sometimes used as a rebuke to children who are too free with their pronouns.
Father: "We're almost out of food..."
Boy: "She's gone to the store."
Father: "Who's she? The cat's mother?"
Confusing, eh? (Well, to us Americans, anyway.) Apparently in much of the English speaking world, it is impolite to use informal pronouns with people to whom one should show respect -- one's mother, father, aunts and uncles, grandparents, etc. The boy should have said something along the lines of "Mother has gone to the store", or even "Mum's out shopping". But he should not use 'she' to refer to his mother. (My friends and I have never heard of this, so I assume this is not done in America. If any of you Americans have something like this where you live, let me know.)
And where did the cat come from? Well, a male cat is a tom; a female cat is a she, or more often, a she-cat. So, 'she' is for cats. (And Not Your Mother!)
Of course, sometimes one must use the word 'she'; for example, anaphoric uses are okay: "Mom went to the store; she'll be back soon".
Being an idiom, it has a slew of minor variations. It is most often found in its short form; "She's the cat's mother!" It's also altered to refer to different people to whom one may have insulted. "She's the cat's grandmother" or "She's the cat's aunt". It is also often said in two parts -- "Who's she? The cat's mother!" or "It's not she, she's the cat's mother!"
Apparently this is falling out of usage, as the world mellows. This is something you're more likely to hear from a grandparent than a mother, although I personally think it would be great to annoy siblings and nephews with.
And Bob's yer uncle!
dcaine says: I wanted to comment on use of "she's the cat's mother. My mother was born in 1903, Maryland, USA and used this phrase. My mother got it from my grandmother, born 1881 in same place.
And a tip o' the hat to Albert Herring, who cleared up some points on proper usage.