A translation circuit, or translation matrix, was a part of a TARDIS that allowed for the instantaneous translation of most languages spoken or written in the universe.
Basic functionality
As the Ninth Doctor once told Rose Tyler, the translation circuit was a "gift of the TARDIS, a telepathic field that gets inside your brain — translates." (TV: The End of the World) The Doctor himself was a part of the circuit, without whom the circuit was broken. When the Tenth Doctor was incapacitated due to post-regenerative crisis, for example, Rose lost access to the circuit, and could no longer understand languages she did not personally know. (TV: The Christmas Invasion) Sometimes, the translation circuits would not work for passengers in different timezones. When Nyssa was trapped in 1957 while the Fifth Doctor and the TARDIS were in 1963, German signs were not translated for her. (AUDIO: 1963: Fanfare for the Common Men) Bernice Summerfield experienced the effects of the translation circuit only when the TARDIS was around. As an archaeologist, she always felt this was "cheating". (AUDIO: Lies in Ruins)
Exactly who could take advantage of the circuit, and under what circumstances, was variable, suggesting the Doctor could change its settings in some way. The Fourth Doctor told Sarah Jane in Renaissance Italy that it was a "gift of the Time Lord" that he allowed her to share. (TV: The Masque of Mandragora) Bernice Summerfield found that she could understand languages such as Hindi and Sontaran and mused upon this in her diary. (PROSE: All-Consuming Fire)
I hadn't had to learn a word: somehow my association with the Doctor had enabled me to understand any language I came across.
During the incident with the Sycorax on Christmas 2006, the translation circuit worked for all humans on the Sycorax ship, even though the Tenth Doctor was not, at the time, apparently aware of their presence. (TV: The Christmas Invasion) Even when Davros had swapped minds with the Doctor, he was able to 'share' his body's gift of translation with Flip Jackson and Jared Ramon when they arrived back in 1815, allowing them to understand French despite the fact that Flip and Jared had never been inside the TARDIS. (AUDIO: The Curse of Davros) On yet another occasion, the Seventh Doctor told Elizabeth Klein that the telepathic field was limited to a certain radius around the TARDIS, with the Doctor losing the ability to communicate with the native Vrill once he moved beyond that radius, although this can be attributed to the fact that the Vrill communicated via smell, with the result that the language was so complex that the Doctor would be unable to even attempt to speak it on his own. (AUDIO: Survival of the Fittest)
The Ninth Doctor once mentioned that the translation systems had a swear filter that prevented the passengers from hearing any swear words; even an angry cavewoman's rants were translated as her saying, "Blinking", rather than a more apt phrase. (PROSE: Only Human)
The translation circuit was apparently capable of translating for creatures without a language. Marnal, the previous owner of the Doctor's TARDIS, identified the animal language translators as one of the many functions in the TARDIS that was no longer in operation after the Doctor's botched repair work on his ship, but the Eighth Doctor dismissed these as irrelevant. (PROSE: The Gallifrey Chronicles) However, it would appear that the Doctor would later at least attempt to repair these circuits, as the Eleventh Doctor was able to barely communicate with a Krafayis, which were only of animal-level intelligence, (TV: Vincent and the Doctor) and later apparently understood a horse when it told him its name. (TV: A Town Called Mercy)
Multilingual individuals were sometimes capable of identifying what language was being spoken. When the Fourth Doctor spent time conversing with Emmeline Neuberger, a native German speaker who also spoke reasonably good English, he responded to her with equal ease when she spoke to him in English and German without showing any sign that he noticed her change in language, Emmeline noting that she couldn't tell what language he was addressing her in. (PROSE: Wolfsbane) The Doctor tended to have more luck in identification. The Fourth Doctor identified the mercenary trader Garron as originating from Somerset simply by hearing his accent, despite the fact that he was on the planet Ribos at a time before it had become aware of other worlds and hence would have no reason for an Earth native to visit it. (TV: The Ribos Operation) On another occasion, the Tenth Doctor used Madame de Pompadour's particular French accent to identify the century in which she lived, while Mickey simply assumed she was speaking English. (TV: The Girl in the Fireplace) When Rose was talking to a pair of African mercenaries, she continually heard them talking in English even when they explicitly said that they would switch to an obscure African dialect that was only known in a small part of the continent in the 22nd century. (PROSE: The Art of Destruction)
The TARDIS translations were closer to the original language than other translators, although it would allow 'local' translations to take over if languages were already being translated. Selachian battlesuits had built in translators, giving them a harsh voice. When removed from their armour and in range of the TARDIS, their speech sounded more melodic, closer to the Ockoran's natural song-like language. (PROSE: The Final Sanction) Likewise, the translation circuit did not compensate for the glitch in Sil's translation system that caused his voice to sound so disturbing. (TV: Vengeance on Varos)
The TARDIS translator gave some individuals noticeable accents. It made Elizabeth Klein, a German-speaker, perceive the Seventh Doctor as having a "stuffy Prussian accent", (AUDIO: Survival of the Fittest) where he traditionally sounded Scottish to most native English speakers. Vincent van Gogh, a native Dutch speaker, assumed that Amy was from Holland due to her having a similar accent to his own. (TV: Vincent and the Doctor) However, the Second Doctor, Jamie, Ben, and Polly still spoke in their usual accents while speaking Russian. (AUDIO: The Night Witches)
It has been suggested on some occasions that the TARDIS translation also slightly modified peoples' minds so that they did not consciously register that they should be unable to understand the languages they were listening to. The Doctor was able to deduce that Sarah had been hypnotised when she directly asked how she was able to understand Italian during a trip to the fifteenth century. (TV: The Masque of Mandragora) On another occasion, when the Brigadier asked how he was able to understand German during a trip to a party thrown by the Nazis in Hitler's honour in 1942, the Sixth Doctor reflected that the champagne the Brigadier had drunk was probably responsible for him asking the question. (PROSE: The Shadow in the Glass)
As noted by Bill Potts, the circuit also appears to handle lip synching, in that the speaker being translated will also appear to be moving their lips in time with what is being said in translation. (TV: The Eaters of Light)
Failure to translate
Although the circuit could translate a vast majority of languages that it encountered in the universe, there were some instances where the circuit failed.
Reasons for this varied, including debility of the Doctor, (TV: The Christmas Invasion) age of the language, (TV: The Impossible Planet, AUDIO: Absolute Power) and complexity or simplicity of the language. (AUDIO: Judoon in Chains, Death and the Queen) [additional sources needed]
In his second incarnation, the circuit failed to translate both German and French while on a planet controlled by the War Lords, while it did translate the War Lords themselves. (TV: The War Games) Whilst visiting Vincent van Gogh, the Doctor's TARDIS was covered in posters which remained in French. (TV: Vincent and the Doctor) Of course, it was established that there was a lag in translation of the written word. (TV: A Good Man Goes to War)
In his third incarnation, the Doctor knew the eyebrow language of the Delphon- musing later that this was one language that could not be translated automatically and it had taken some time for him to learn the subtle nuances of the language- but did not understand Chicken or Polari. (TV: Spearhead from Space, Carnival of Monsters, AUDIO: The Blame Game, The Havoc of Empires) When he was on the island of Salutua, the Doctor was unable to communicate with the Semquess, but this can be attributed to his current distance from the TARDIS- which was back in England thirty-six years in the future- and the complexity of the Semquess language, which both featured underwater communication and use of their multiple limbs. (PROSE: The Eye of the Giant)
When the Fifth Doctor and Vislor Turlough initially encountered the Phiadoran/Vrall Lytalia on the Moon in 1878, Turlough assumed that his inability to understand her was natural, but the Doctor soon realised that they had only failed to understand her because her language syntax was being replaced by English based on knowledge extracted from the brain enzymes of Sub-Lieutenant Granby, and she had merely been making random sounds to conceal this until she had another excuse to speak English. (PROSE: Imperial Moon)
On Heaven, the Seventh Doctor claimed that the TARDIS could not translate writing since it didn't have a mind to read, and thus he was unable to read Heavenite. (PROSE: Love and War)
The Hervoken language, encountered by the Tenth Doctor and Martha, couldn't be fully translated, though the essence was understood. (PROSE: Forever Autumn)
Even some Earth languages did not translate. In the time of the Fifth Doctor, the TARDIS was unable to translate for Kurkutji, who spoke an old Aborigine dialect, although the Doctor's companion Tegan Jovanka was able to speak the language normally through personal knowledge. (TV: Four to Doomsday) The Seventh Doctor noted that the TARDIS translator had always had trouble with Esperanto. (AUDIO: A Thousand Tiny Wings)
Some languages that were too complex, like the Sittuun language encountered by Amy and the Eleventh Doctor, would not be translated. (PROSE: Night of the Humans) In other instances, if the idea was too complex, translation would be incomplete or faulty, an example being when the Fifth Doctor used the phrase, "We will have been here before", to describe the TARDIS arriving at a crater on the Moon in the early twenty-first century when the TARDIS would visit that location in 1878 in its personal future. (PROSE: Imperial Moon) The Seventh Doctor and Mel encountered Golosian, which was also too complex to be translated by the TARDIS, although a translator created during their visit to Dark Space 8 was able to provide an accurate translation of the Golosian language. (AUDIO: Bang-Bang-a-Boom!) The Foamasi language also went untranslated. (TV: The Leisure Hive) This was apparently due to the complexity, as the language stimulated the visual cortex of the brain, meaning Foamasi effectively saw their language. (PROSE: Sleepy) On the other hand, Judoonese was too basic to be translated. (AUDIO: Judoon in Chains)
Even later, languages which predated the universe could not be translated. The script written on the planet Krop Tor was not translatable. (TV: The Impossible Planet) A very old minotaur-like species couldn't be translated properly. Its words sounded like roars and grunts to the companions, and the Eleventh Doctor had difficulty understanding several words. (TV: The God Complex) Inscriptions on the very ancient Cradle of the Gods weren't translated in the eyes of the Eleventh Doctor and his companions. (PROSE: The Dalek Generation)
The Doctor was able to bypass the translation circuits to say minor phrases in other languages, such as "thanks" in Vietnamese (PROSE: Interesting Times) or "Allons-y". (TV: Army of Ghosts, et. al.) The Twelfth Doctor once claimed this was because of his precise control over the translation circuits. (COMIC: Four Doctors)
As Clara found out, while her speech translated into perfect Russian, phrases like "hen night" and "karaoke" did not translate. (TV: Cold War)
Sabotage
On some occasions, the TARDIS translation circuits could fail to work through deliberate action on the part of others. On one occasion, the Master disrupted the Third Doctor's attempt to talk to him by hacking the translation and telepathic circuits of the Doctor's TARDIS and feeding the Doctor's words back to him before he had spoken, resulting in his words coming out backwards. (TV: The Time Monster)
Translation could be affected by the listener's own perceptions, as when the circuit initially failed to allow Anji Kapoor to understand the language of the intelligent tigers of the planet Hitchemus because Anji couldn't accept that the Tigers were sentient. (PROSE: The Year of Intelligent Tigers)
When facing the alien artist Monos, who sought to use a psychic gauntlet to imprint his identity across London, the Doctor was able to defeat him by having Amy and Rory deliberately sabotage the translation circuit, scrambling all language in the ship's vicinity so that nobody could process Monos's name properly, giving the Doctor time to disable his equipment. (COMIC: Sticks and Stones)
User interaction with the circuit
Companions
When the Tenth Doctor and Donna arrived in Pompeii, Donna wondered what would happen if she spoke Latin to the locals, whose native language was Latin. Instead of hearing Latin, at least one native speaker of Latin believed her to be speaking Celtic. (TV: The Fires of Pompeii)
Most of the Doctor's companions were native speakers of English, and the Doctor himself was a "master of English" — indeed a fan of the language. (AUDIO: ...ish) However, he did occasionally travel with non-English-speakers. Elizabeth Klein, a native German speaker, required the circuit just to communicate with the Doctor. (AUDIO: Survival of the Fittest)
The circuit could apparently be influenced by the passengers who travelled in it. When the Eighth Doctor, Fitz and Anji landed on a world where they encountered the "mooncalfs" - people born with some kind of genetic deformity, regarded as an abomination in this world - Fitz recalled the term from his childhood as referring to someone who was either a bit slow or someone considered a freak, speculating that the TARDIS translator chose a word that he knew to describe them because it was more "tuned in" to his "wavelength" given that he had spent longer travelling with the Doctor than Anji. (PROSE: Vanishing Point)
Anji had a particularly unusual relationship with the translation circuit. When dealing with the sentient tigers of the planet Hitchemus, she was actually able to ignore the circuit's attempts to translate at first because she initially couldn't accept that the tigers were sentient. (PROSE: The Year of Intelligent Tigers) She actually learned a new language whilst travelling with the Doctor. Caught in the midst of the Spanish Civil War, she used her pre-existing fluency in French to quickly assimilate Catalan while she was in Barcelona. (PROSE: History 101) It was unknown how she was able to experience traditional language learning whilst still nominally linked to the TARDIS.
Before travelling with the Eighth Doctor, Mary Shelley had been trying to learn German. When she travelled to Austria with the Doctor, she assumed that she was just picking up German faster when she could easily read the newspaper. (AUDIO: The Silver Turk)
When travelling with Jago and Litefoot, Litefoot immediately noticed that the Venusians were speaking English when they should be speaking an alien language. Jago, on the other hand, thought Venusians would naturally speak English. (AUDIO: Voyage to Venus)
When Troy Game first travelled in Lord Roche's TARDIS, as she was already naturally capable of speaking other languages due to her race's innate telepathy, the TARDIS instead augmented her natural abilities, increasing her telepathy and granting her a degree of empathy. (PROSE: The Suns of Caresh)
Before joining the Tenth Doctor in the TARDIS, June was unable to understand a conversation held in an alien language between the Doctor and Bartholomew, an alien. The Doctor, being naturally fluent in English, was able to speak English and the language of the alien simultaneously so that June could understand him. (PROSE: The Slitheen Excursion)
Behind the scenes
The TARDIS crew's ability to understand other languages wasn't explained in any story in any medium until The Masque of Mandragora. Even so, that was the only story in the whole of the 1963 version of Doctor Who to even touch upon the subject of language translation. The issue has had much greater prominence in the BBC Wales programme, which has then had an impact on the Big Finish past Doctor stories.