It goes EERrwWwoooooSHHHhEERrwWwoooooSHHHhEERrwWwoooooSHHHh.
The "wheezing, groaning" sound of the TARDIS was achieved by sound designer Brian Hodgson taking his mother's front door key and scraping it down the bass strings of an upright piano.
It was then electronically processed into the sound still heard today.
In Underworld (1978), the very similar sound of the TARDIS landing was identified by a spaceship's computer as being the "relative dimensional stabiliser in materialisation mode". The takeoff sound is presumably therefore the "relative dimensional stabiliser in dematerialisation mode".
The "wheezing, groaning" description was first invented by Terrance Dicks, script editor, in one of his many novelisations of the classic stories. Most of the Target novelisations used the same phrase, as did many of the novels that followed the classic series' cancellation. The description has been used by characters in-universe in the novels and in the comics, but (as of series 6) never on TV. Instead, the 11th Doctor attempted to actually make the TARDIS sound with his voice (much like the short version of the answer above).
- ↑ A few writers tried to come up with different descriptions, or used in-jokes, like "the sound of a wheezing, groaning man torturing a piano with his keys" or "the wheezing, groaning sound of an overused metaphor".