Rhyming slang at its simplest replaces one word with another that rhymes with it. Thus "chime" might mean "rhyme". Such slang becomes more clever, and ultimately more impenetrable, when it continues on to rhyme the second word of a two-word phrase with the intended meaning; thus "doorbell chime" might be used to mean "rhymes." And in the final manipulation, even this would be replaced with the first word of the phrase, thus "doorbell" has come to mean "rhyme." This is a hypothetical example, but it gives a fair sense of how such slang works.
There are perhaps four essential elements to rhyming slang. First, there is the use of the interesting new phrase, which demonstrates that one is in the know. Second, there is the choice of a particularly incongruous and/or humorous phrase, demonstrating that one is clever -- so long as one's audience can figure out the joke, but only barely, this makes such slang funny. Third, of course, there is the sheer bloody-minded desire not to be understood by those unlike you; after all, if you are in the habit of referring to poetry as having a lot of "doorbells," nobody is likely to have the remotest idea of what you are talking about. Finally, as you will see, a lot of this is really about being able to talk dirty without quite seeming to, which of course also makes the dirty jokes dirtier and funnier.
Here's an example:
'Allo me old china - wot say we pop round the Jack. I'll stand you a pig and you can rabbit on about your teapots. We can 'ave some loop and tommy and be off before the dickory hits twelve.
or, to translate
Hello my old mate (china plate) - what do you say we pop around to the bar (Jack Tar). I'll buy you a beer (pig's ear) and you can talk (rabbit and pork) about your kids (teapot lids). We can have some soup (loop de loop) and supper (Tommy Tucker) and be gone before the clock (hickory dickory dock) strikes twelve.
I’m ‘aving terrible trouble with me April [How can such a simple word have so many convoluted references? April in Paris - Aris (from Aristotle - bottle which is from bottle and glass - arse.) Whew – Thanks to Peter Chrisp]
Arse
Aristotle=Bottle=Bottle and Glass=Arse; therefore, Aris=Arse
I gave him a good kick up the Aris [Thanks to Ray Davis.] See also bottle.
Arse
Bottle and Glass
I gave him a good kick up the bottle.
Arse
Khyber Pass
Stick it up your khyber.
Arse
Rolf Harris
She kicked him in the Rolf [Rolf Harris wrote "My Boomerang Won't Come Back". See the reference above to Aristotle. Thanks to Matt Fisher] {Obviously a recent one-- ed.}
Arsehole
Elephant & Castle
He's a bit of an elephant [Thanks to Steve Fuller]
Arsehole
Jam Roll
That geezer is a right jam roll. [Thanks to Robert Lynch]
Arsehole
Merry Old Soul
‘e’s a bit of a merry old soul [Thanks to Sanor]
Aunt
Mrs. Chant
He didn't know what to get his Mrs. Chant for Christmas [Thanks to Alan Little]
Mayhew on Rhyming Slang
". . there is the ‘Cadgers’ (beggars’) cant’, as it is called—a style of language which is distinct from the slang of the thieves, being arranged on the principle of using words that are similar in sound to the ordinary expressions for the same idea. ‘S’pose now, your honour,’ said a ‘shallow cove’, who was giving us a lesson in the St. Giles’ classics, ‘I wanted to ask a codger to come and have a glass of rum with me, and smoke a pipe of baccer over a game of cards with some blokes at home—I should say, Splodger, will you have a Jack-surpass of finger-and-thumb, and blow your yard of tripe of nosey-me-knacker, while we have a touch of the broads with some other heaps of coke at my drum?’"
HenryMayhew and John Binny, The Criminal Prisons of London, 1862
Links
Cockney Rhyming Slang -- "the world's greatest dick'n'arry [dictionary, got it?] of Cockney Rhyming Slang," and I can well believe it. I particularly like Austin --> Austin Power --> shower, as in "Just off to jump in the Austin."