Real world interface

The real world interface, on some occasions referred to as the door, was the part of a TARDIS which connected the outer plasmic shell of the ship to the interior. (TV: Castrovalva) The First Doctor's TARDIS had two doors, the outer door and the inner door. (COMIC: The Secrets of the TARDIS)

Any Matrix interface outside of the Matrix itself was also referred to as a real world interface. (AUDIO: Ascension)

Functions
By crossing the dimensional threshold of the interface a person or object underwent an instantaneous translation from the ongoing space-time continuum occurring outside the vessel to that pertaining in the separate micro-environment within (or vice versa). This process was known as "going into the TARDIS" or "coming out of the TARDIS". (TV: Logopolis)

Opening the doors while the Doctor's Type 40 was in transit usually had grave consequences, (TV: Planet of Giants, Warriors' Gate) but once materialised, the TARDIS force field usually protected the interior even if the doors were opened to a hostile environment. For example, if the doors were opened in a vacuum, the interior of the ship would not undergo explosive decompression. (TV: Four to Doomsday, The Runaway Bride, The Stolen Earth)

On the other hand, the outside environment could affect the opening mechanism. On the ice planet Calufrax Major, it was so cold that the doors did not open without a de-icer. (PROSE: The Secret in Vault 13)

Real world interface of the Doctor's TARDIS
The Doctor's TARDIS originally had what appeared to be two sets of doors, the police box doors of the outer shell, which were manually operated from within and without, and a pair of more substantial doors covered on both sides in the usual roundel motif. The inner doors were usually power-operated — operating a switch on the console or opening the outer doors manually caused the inner doors to open automatically. A dark void linked the two sets of doors. The pressure exerted by the door servos was not especially great; a Cyberman was able to force the doors open against the mechanism. In the event of a total power failure, the doors could be cranked open and shut manually. (TV: The Claws of Axos, Death to the Daleks, Snakedance, Earthshock, PROSE: Full Circle)

By the time he met Rose Tyler in his ninth incarnation, the Doctor's TARDIS had a simpler arrangement, with the police box doors of the ship opening directly onto the console room. The doors were usually manually operated during that era. Despite the appearance of simplicity, the Doctor believed the protection provided by the doors when closed to be exceptionally strong, telling Rose that "the assembled hordes of Genghis Khan couldn't get through that door — and believe me, they've tried." (TV: Rose) Queen Elizabeth I once pointed out that while the Tenth Doctor's TARDIS was bigger on the inside, the door wasn't, and her head was nearly taken off when the Doctor rode out of the TARDIS with Elizabeth on a Zygon which took on the body-print of a horse. (TV: The Day of the Doctor) On one occasion, when the TARDIS' exterior shell shrank, so did the door on the interior. (TV: Flatline)

River Song revealed that the future Doctor she knew habitually opened the TARDIS doors by clicking his fingers, something the Tenth Doctor said was impossible. However, shortly after her death, he tried this and found it worked. (TV: Forest of the Dead) He did not routinely use this ability in the rest of this incarnation, although his eleventh incarnation did later demonstrate it to Amy Pond. (TV: The Eleventh Hour)

During the time of the Eleventh, Twelfth, and Thirteenth Doctors, the TARDIS doors could still be opened and closed remotely from the central console. (TV: Journey to the Centre of the TARDIS, Flatline, Resolution)

The real world interface could vanish entirely with no way into or out of the TARDIS once the TARDIS entered siege mode. This could not be circumvented manually, and required additional power to "turn it off". (TV: Flatline)

Flux event
During the Flux event, the Thirteenth Doctor's TARDIS began spawning new doors in the interior each time the Doctor and her companions entered, though, at first, the outer door remained unchanged.

When events began changing at the Temple of Atropos, with the arrival of Yasmin Khan and Vinder, the outer doors vanished entirely, so that each side of the police box exterior appeared the same. (TV: War of the Sontarans) This dimensional interference later affected the Williamson Tunnels, as well, producing entirely too many doors to different places. (TV: The Vanquishers)

After defeating and Azure and restoring the Web of Time, the Thirteenth Doctor needed to recalibrate the TARDIS in order to remove these interferences. Afterwards, normal use of the TARDIS doors continued. (TV: Eve of the Daleks)