The Doctor's TARDIS — sometimes called the Ship by the First Doctor, and most often known simply as the TARDIS — was the Doctor's primary means of transport. It was capable of travelling through space and time. The Doctor voyaged in his vessel from the Big Bang (TV: Terminus, Castrovalva, AUDIO: Slipback) to the end of the universe in the year 100 trillion. (TV: Utopia).
Other Time Lords frequently characterised the Doctor's TARDIS as woefully out-of-date. (TV: The Time Meddler, The Claws of Axos, The Ribos Operation) Indeed, by at least the time of the Doctor's fourth incarnation, if not much earlier, the model — called a "Type 40" — had been pulled from general service on Gallifrey. (TV: The Deadly Assassin, The Invasion of Time)
The craft was prone to a number of technical faults, ranging from depleted resources (TV: An Unearthly Child, The Wheel in Space, Vengeance on Varos) to malfunctioning controls (TV: The Edge of Destruction) to a simple inability to arrive at the proper time or location. (TV: The Visitation, Attack of the Cybermen, The Eleventh Hour, Victory of the Daleks, The Girl Who Waited and many others) However, because the TARDIS was a living being, these "faults" may instead have been at least partially attributed to the manifestation of the Ship's free will. Indeed, the TARDIS herself once told the Eleventh Doctor that she may not have always taken him where he wanted to go, she did take him where he needed to be. (TV: The Doctor's Wife).
In his first incarnation, the Doctor implied he had built his TARDIS himself. (TV: The Chase) Whether there was some truth to this or not, he and others later stated that he had, in fact, stolen it. (TV: The War Games, Logopolis, Planet of the Dead et. al.) The TARDIS was once owned by the Time Lord, Marnal, and due to his exile at the time of the Doctor's procurement of the TARDIS, he claimed that the Doctor stole it. (PROSE: The Gallifrey Chronicles) Other accounts differ to this, and implied it came from the general, government-controlled "stockpile" of TARDISes after the model had been officially decommissioned. (TV: The Deadly Assassin, PROSE: The Exiles, COMIC: Time & Time Again) The Fourth Doctor told Adric that "it was in for repairs on Gallifrey when [he] borrowed it". When the Alzarian countered that he thought the Doctor outright owned the vehicle, the Doctor said, "Well, on sort of 'finders, keepers' basis, yes." (TV: Logopolis.
When the Doctor first decided to leave Gallifrey, he had the chance to take a Type 53, but dismissed it as "soulless" in favour of the Type 40. (PROSE: Lungbarrow) The TARDIS herself said she was "a museum piece", though this may have been figurative. (TV: The Doctor's Wife).
None of these accounts precluded the possibility that he had somehow been responsible for its creation. Indeed, another account compromised between theft and creation. It claimed while the Doctor had not built the TARDIS from scratch, he had substantially modified/rebuilt it. According to this view he achieved control of the TARDIS without using a direct mental link. This let him bypass the feature on most TARDISes which sent a tracking signal to the Time Lords. (PROSE: The Taking of Planet 5).
These accounts notwithstanding, the most direct commentary on the Doctor's acquisition of the TARDIS came from the TARDIS herself. When House transferred the soul of the TARDIS into Idris, the TARDIS gave her side of the story. She confirmed she had been out of commission, a "museum piece", when the First Doctor met her. She also confirmed that the Doctor had stolen her, denying the Eleventh Doctor's attempt to characterise the action as "borrowing". She also stated that she had stolen him, and had no intention of ever giving him back. She was unlocked and had deliberately let him steal her because she wanted to explore the universe and sensed he would be an ideal match. (TV: The Doctor's Wife) According to the TARDIS, the Doctor's first words to her, some seven hundred years before, were:
You are the most beautiful thing I've ever known.
The precise model number of the Doctor's TARDIS was a matter of some confusion, particularly when it was compared to those of other Time Lords. The Monk claimed to have a Mark IV TARDIS, while the Doctor had a Mark I. (TV: The Time Meddler) The dematerialisation circuit of the Master's TARDIS was a Mark II, compared to the Doctor's Mark I. (TV: Terror of the Autons) However, it was unclear whether this meant that the Master's TARDIS, as a whole, was a Mark II. When the Teselecta scanned the Doctor's TARDIS, its records stated the timeship was a Type 40, Mark 3. (TV: Let's Kill Hitler)
During a visit by the Fourth Doctor to Gallifrey, the Doctor's TARDIS was unambiguously called a "Type 40". At that time, it was made clear that all other Type 40s had long since been officially decommissioned and replaced by newer models. The fact that the Doctor's TARDIS was a Type 40 was not common knowledge, even to the Castellan. (TV: The Deadly Assassin) This designation was used with greater frequency afterward. It was even used by the Eleventh Doctor as an excuse to Winston Churchill for his tardy response to Churchill's summons. (TV: Victory of the Daleks) When the TARDIS had the opportunity to speak to the Eleventh Doctor in the body of Idris, she called herself a "Type 40" without any qualification. (TV: The Doctor's Wife)
The Eleventh Doctor appeared to know his TARDIS' date of manufacture, as one chronicle indicated that he celebrated "her birthday". (PROSE: Dark Horizons).
Almost all TARDISes were designed to blend into their surroundings by means of a mechanism usually called the "chameleon circuit", but occasionally the "camouflage unit". Some later models seemed to let the pilot choose a desired exterior, overriding what would have been "natural" for the surroundings. (TV: Time and the Rani, Time-Flight)
The Doctor's TARDIS would have had both abilities, were the chameleon circuit operational. Before he met Ian and Barbara, the First Doctor had landed on Iwa, where the TARDIS had posed as a boulder in that planet's desert. (PROSE: Frayed) On Quinnis, the First Doctor was unhappy when the TARDIS landed in a bazaar and chose to turn into a market stall, complete with a striped awning. (AUDIO: Quinnis) The Fourth Doctor showed Adric how the TARDIS could be changed to the shape of an Egyptian pyramid, implying he could override the chameleon circuit's "automatic" functionality. (TV: Logopolis)
In any case, the defining characteristic of the Doctor's TARDIS was that its chameleon circuit had broken after assuming the shape of a police box in 1963 London. It had been working until it landed in I.M. Foreman's junkyard. The Doctor's granddaughter said the TARDIS had previously appeared as a sedan chair and an ionic column. The Doctor and she expressed surprise that it had not changed form when they arrived at a new destination. (TV: The Cave of Skulls)
By his eleventh self, the Doctor was telling his companions that the chameleon circuit was working, but due to a fault, invariably assumed its customary police box shape:
It’s camouflaged. It’s disguised as a police telephone box from 1963. Every time the TARDIS materialises in a new location, within the first nanosecond of landing, it analyses its surroundings, calculates a twelve-dimensional data map of everything within a thousand-mile radius and then determines which outer shell would best blend in with the environment.... and then it disguises itself as a police telephone box from 1963.