TARDIS as almost everyone knows nowadays stands for Time and Relative Dimension in Space. Or sometimes DimensionS in Space if the script editor isn't watching.
In the VERY FIRST episode of Doctor Who, called "An Unearthly Child", Dr Who's granddaughter Susan who is the unearthly child of the title, tries to explain his time machine, inside of which they are, to her BEWILDERED teachers, Ian and Barbara:
"I made up the name from the initials… I thought you would understand when you saw the different dimensions inside from outside."
You can see why they call her UNEARTHLY can't you!
This is all very COOL and ENIGMATIC in the opening episode, but it does – with HINDSIGHT – set up a little bit of a PROBLEM for the continuity cops.
If Susan came up with the name "TARDIS" then that implies that the timeship is a relatively RECENT invention, and that Dr Who probably had something to do with inventing it.
This is pretty much the premise of the MOVIE version of "DOCTOR WHO AND THE DALEKS" in which the Doctor is a wacky professor but entirely HUMAN and has invented time travel in a hut in his back garden.
However, future adventures will go on to explain that in fact Dr Who comes from a VERY, VERY old civilisation and that they have had time travel for a VERY, VERY long time, it having been invented about ten MILLION years ago by Omega and Rassilon and the Other (canonicity pending).
So HOW COME Susan could come up with the name TARDIS if she is REALLY Dr Who's granddaughter AND he is less than four-hundred-and-fifty years old (according to Mr Pat when he is the second Dr Who)?
The New Adventure "LUNGBARROW" – which is a JOLLY GOOD READ, even if it is a bit frightfully expensive to get a paper version from eBay – goes to very CONVOLUTED lengths to explain.
Susan is REALLY the granddaughter of this chap called "the Other", who was the third person who lived long ago in history and helped set up the Time Lords. His real name has been forgotten, probably because Rassilon airbrushed him out of the history books, so I shall call him Fred. Since Fred was there at the design breakfast meetings, he was able to have a say in the naming of the new time machines, and his cute little button of a granddaughter (that would be Susan, remember) gave him the idea of calling them TARDISes.
What really happened to Fred is that in order to get away from Rassilon – who was bugging him – he chucked himself into the new baby-Time-Lord-making machine that he and Rassilon had invented to get over the slight snag of the entire planet being cursed to perpetual sterility by the last of the Pythia priestess-queens. (Er, that reads better in long form!)
Anyway, ten million years later and the baby-machine clunks out a new Time Lord who just HAPPENS to have most of Fred's DNA. And he's Dr Who. Then Dr Who gets taken back in time all the way to the start of Time Lord history again and meets Susan and she thinks that he's Fred only bopped on the head and a bit confused, so they both trip off merrily in the TARDIS and have some dashed queer adventures in the French Revolution and the planet Quinnis in the fourth universe and Coal Hill School, Shoreditch.
All of which is a terrifically complicated way of ENTIRELY MISSING THE POINT.
TARDIS is NOT what the Time Lords call their time machines; it is what Susan calls Dr Who's time machine. In the same way that my Daddy Richard calls his car Penfold, TARDIS is Susan's PET NAME for their timeship.
To start with, Dr Who himself mostly calls the ship "the ship". Sometimes (although this is rather more in that movie version) he and Susan even refer to it as "TARDIS" rather than "the TARDIS", just like sometimes Captain Quirk calls his Constitution Class heavy cruiser "Enterprise" and sometimes "the Enterprise".
The first time that we encounter ANOTHER timeship, it is VICKY – Dr Who's young friend who does not know better – who says "the Monk has got a TARDIS".
The MAIN Time Lord story for reference is – isn't it always – "THE DEADLY ASSASSIN" and in that story it is quite explicit that the Time Lords themselves refer to their timeships as "TT capsules". (Time Travel Capsules, probably.)
So Dr Who's time machine is a Gallifreyan Type Forty TT Capsule that is CALLED "TARDIS" or "the TARDIS" by her crew.
Obviously this is flatly contradicted by any number of stories – usually with ludicrous loon, the Master in them – when he refers to Dr Who's "TARDIS" or Dr Who refers to his "TARDIS" because the script editor thinks that "TARDIS" means "car" and not "Penfold".
So you are stuck with the COMPLICATED explanation.
Still, these days the Time Lords have all gone foom and there are no other TT Capsules left so "TARDIS" and "timeship" really are synonymous again.