Chameleon circuit

The chameleon circuit was a component of a TARDIS which enabled its outer plasmic shell to assume any shape, in order to blend in with its surroundings.

General uses
A TARDIS with a functioning chameleon circuit could appear as almost anything, if its owner so desired. The owner would be able to programme the circuit to make it assume a specific shape, or else the TARDIS itself could decide what form to take. The chameleon circuit would then ideally make the TARDIS fit in to the natural environment of a specific destination. The perception filter also helped a TARDIS to disguise itself. (TW: Everything Changes)

The Doctor once referred to the chameleon circuit of his TARDIS as a "cloaking device". The fact that he hesitated slightly before using the term "cloaking device" suggests that he was trying to think of an alternative to "chameleon circuit". (DW: Doctor Who)

History
The chameleon circuit of the Doctor's TARDIS, however, rarely worked properly since the First Doctor made a sudden departure from Totter's Lane, Shoreditch, 1963. Subsequently, the TARDIS usually remained in the shape of a London police box. (DW: An Unearthly Child)

The Doctor hoped to repair it in Logopolis by using Block Transfer Computations when the Master interfered with the Logopolitans' calculations. (DW: Logopolis) He succeeded in repairing it for a brief period when he returned to Totter's Lane in 1986, but after it began to transform into shapes that still refused to blend into their surroundings - and on some occasions even made it hard to figure out how he was meant to enter his ship in the first place - he reverted it back to its usual police box form. (DW: Attack of the Cybermen)

During his seventh incarnation, the Doctor briefly enabled his ship to work again (NA: Conundrum), but reset it back to a police box after Mortimus hacked into the circuit and nearly gave away its location by turning it into the Statue of Liberty while he was materialised around Nelson's Column. (NA: No Future)

It should be noted that, in the above example, the Doctor was travelling in a TARDIS he had acquired from an alternate timeline where his third incarnation had been murdered (NA: Blood Heat).

After Donna Noble reported her encounter with Rose Tyler to the Doctor, he began noticing that the words "Bad Wolf" had begun to appear everywhere, including replacing the traditional police box lettering on his TARDIS. These words were also visible from the interior of the TARDIS over the doorway. It is possible the chameleon circuit was somehow activated in order to make this change. (DW: Turn Left)

Later, when Donna briefly had a Time Lord consciousness, she began to tell the Doctor how to repair the circuit, but her brain began to overload before she could complete the instructions while stating "Binary" in a repetitive fashion. (DW: Journey's End)

However, later incarnations of the Doctor showed no particular interest in repairing the circuit, with the ninth incarnation more or less telling Rose that he liked its appearance. (DW: Boom Town) When the Master took possession of the TARDIS for an extended period of time, he also declined to fix the chameleon circuit (although the fact that he was incapable of travelling anywhere between two locations may have meant that he simply saw no point in it). (DW: The Sound of Drums / Last of the Time Lords)

When the circuit was not in operation the TARDIS exterior appeared as a plain, grey cabinet which opened via a sliding door. (DW: The War Games)

However, when the TARDIS regenerated after crashing into Amy Pond's backyard, it had taken the form of a slightly different blue police box (DW: The Eleventh Hour). It is possible that the Chameleon Circuit actually works, but the doctor chooses to leave it as a police box.

The Monk's TARDIS
The Monk's TARDIS appeared as a sarcophagus in an English church of 1066, (DW: The Time Meddler), a large stone on Tigus, and a stone block in Ancient Egypt. The Doctor caused it to appear as an Ionic column, a stage coach, a tree, an igloo, a rocket, and a bi-plane before impishly setting it as a police box to distract the Daleks. (DW: The Daleks' Master Plan)

In 1976 London it assumed the form of a wooden desk. (NA: No Future)

The Master's TARDIS
The TARDIS belonging to the Master took on several different forms during his many encounters with the Doctor. These included:
 * A horse box (DW: Terror of the Autons),
 * A bank of computers(DW: The Time Monster),
 * A Corinthian column (DW: The Time Monster, Logopolis, Time-Flight),
 * A grandfather clock (DW: The Deadly Assassin, The Keeper of Traken),
 * The calcified Melkur (DW: The Keeper of Traken),
 * A police box identical to the Doctor's TARDIS (DW: Logopolis),
 * A fireplace (DW: Castrovalva),
 * The Concorde (DW: Time-Flight)
 * An "iron maiden" (DW: The King's Demons)
 * A space battle cruiser (Birth of a Renegade)

Others
The Rani's TARDIS took the form of both a cabinet (DW: The Mark of the Rani) and a translucent pyramid (DW: Time and the Rani).

Iris Wildthyme's TARDIS took the shape of a red double decker bus (Number 22 to Putney Common), and at one point, according to Iris, was slightly smaller inside than out. (EDA: The Scarlet Empress, The Blue Angel, Mad Dogs and Englishmen, PDA: Verdigris)

The TARDIS of Professor Chronotis looked like his rooms at St. Cedd's College. (DW: Shada)

Behind the scenes

 * Fanon has suggested that when the Doctor "borrowed" his TARDIS from Gallifrey, it was in for repairs, and one of the broken systems was the chameleon circuit.
 * The Chameleon circuit was originally referred to as a "camouflage unit". (DW: The Time Meddler). The next time the device was mentioned, its name was changed to "Chameleon circuit". (DW: Logopolis)
 * The real world reason for the malfunction is thought to be of a far more practical nature: the Chameleon Circuit was intended to allow the TARDIS to blend with its surroundings during the 'historical' episodes which would require an expensive redress of the TARDIS prop for every episode. Others have suggested that the shape was initially selected to provide something that the present audiences would instantly recognise.
 * The Trock band "Chameleon Circuit" are named after this TARDIS component.
 * Owlman's plane in Justice League: Crisis on Two Earth has a device called a Chameleon Circuit that completely cloaks the plane.