Chameleon circuit

The chameleon circuit was the component of a TARDIS which changed its outer plasmic shell to assume a shape which blended in with its surroundings.

Function
Default TARDIS exteriors looked like plain, grey cabinets which opened via a sliding door. (DW: The War Games)

A TARDIS with a functioning chameleon circuit could appear as almost anything desired. The owner could program the circuit to make it assume a specific shape. If no appearance was specified, the TARDIS automatically chose its own shape. (DW: Logopolis)

The chameleon circuit was primarily designed to allow a TARDIS to blend into its surroundings. When a TARDIS materialised in a new location, within the first nanosecond of landing its chameleon circuit analysed the surrounding area, calculated a twelve-dimensional data map of all objects within a thousand-mile radius and then determined which outer shell would best blend in with the environment. (DW: Meanwhile in the TARDIS)

Flaw discovered
After the initial journeys of Ian Chesterton and Barbara Wright, the TARDIS remained fixed in the shape of a London police box. Neither Susan nor her grandfather could explain why the ship had lost its ability to disguise itself. (DW: An Unearthly Child)

Following this initial failure, the Doctor did not show much interest in repairing it. By the time of his ninth incarnation, for instance, he intimated to Rose Tyler that he liked its appearance. (DW: Boom Town) Other Time Lords who encountered the Doctor's TARDIS didn't seem to want to repair it, either. When the Master tried to repair the Doctor's Earth-bound TARDIS, he expressed no particular concern with the fixing the chameleon circuit. (DW: The Claws of Axos). Much later, another incarnation of the Master took possession of the TARDIS for an extended period of time, but made no effort to fix the chameleon circuit. (DW: The Sound of Drums / Last of the Time Lords)

Despite the chameleon circuit being broken, the TARDIS occasionally still changed details of its outer shell's appearance. Though keeping the general form of a police box, it would change such things as the overall exterior size, shade of blue, window panes, lamp colour, and the text which appeared on the telephone hatch. (DW: The Eleventh Hour et al)

Much later, the Eleventh Doctor expounded on the flaw further to Amy Pond, telling her that the circuit scanned the surroundings upon materialisation, decided on the best disguise, but then always disguised itself as a 1963 police box anyway. (DW: Meanwhile in the TARDIS)

Repair attempts
Despite a general lack of enthusiasm for fixing the chameleon circuit, there were nevertheless a few efforts made.

The Fourth Doctor was hoping to repair it in Logopolis by using Block Transfer Computations when the Master interfered with the Logopolitans' calculations. (DW: Logopolis)

Nyssa tried to repair it on her own, without even referencing the TARDIS manual. After double-checking her work and finding it surprisingly sound, the Fifth Doctor chose an overly-ambitious environment for the first test of Nyssa's work. He materialised in one of Earth's oceans, whereupon it turned into a whale - both externally and internally. It became so comfortable in its new form that it seemed to forget that it was a TARDIS at all. Only by transmitting his heartsbeat (sic!) underwater and reminding it of its link to him was the Doctor able to swim into one of its arteries, undo Nyssa's fixes, and return it to its usual police box shape. (BFA: The Deep)

The Sixth Doctor attempt was temporarily successful. He repaired it for a brief period when he returned to Totter's Lane in 1985, but after a few transformations 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 - it reverted back to its usual police box form. (DW: Attack of the Cybermen)

During his seventh incarnation, the Doctor briefly enabled his ship — or, more precisely, a version of his TARDIS (NA: Blood Heat) — to work again. (NA: Conundrum) He later reset it to a police box after the Monk hacked into the circuit and nearly gave away its location by turning it into the Statue of Liberty while it was materialised around Nelson's Column. (NA: No Future)

Later, when Donna Noble briefly had an active Time Lord consciousness, she began to tell the Tenth Doctor how he might be able to repair the circuit, but her brain began to overload before she could complete the instructions. (DW: Journey's End)

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 First Doctor caused it to appear as a motorcycle, an ornamental coach, a covered wagon, a tank and possibly other forms 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, a phone booth, a van, a motorcycle and a filing cabinet. (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. It often adapted to fit a need of the Master or to hide it as a generic object.

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)

Nomenclature

 * Though reference was made to the TARDIS' supposed ability to change to match its surroundings in the first Doctor Who story An Unearthly Child, the chameleon circuit itself was not specifically referred to until The Time Meddler, in which it was called a "camouflage unit." The device was not mentioned again until Logopolis, in which it was called the "chameleon circuit," the term to which it has generally been referred ever since.
 * In the TV Movie, the chameleon circuit was referred to in dialogue as the TARDIS' "cloaking device," a term from Star Trek.
 * The choice of name was inspired by the ability of chameleons to change their skin colour, which has been generally thought to be for the purposes of camouflage.

Malfunction

 * The real-world reason for the chameleon circuit's malfunction is thought to be of a far more practical nature than in-universe: the TARDIS was originally intended to blend with its surroundings during the 'historical' episodes, which would have required an expensive redress or replacement of the TARDIS prop for every story. Others have suggested that the constant police box shape was selected to provide something contemporary audiences would instantly recognise in any story settings.