Translation circuit

A translation circuit was a part of the Doctor's TARDIS that allowed for the instantaneous translation of most languages spoken or written in the universe.

Basic functionality
As the Doctor once told Rose Tyler, the translation circuit was a "gift of the TARDIS, a telepathic field that gets inside your brain — translates." (DW: The End of the World) The Doctor himself was a part of the circuit, without whom the circuit was broken. When he 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. (DW: The Christmas Invasion)

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 Doctor told Sarah Jane in Renaissance Italy that it was a "gift of the Time Lord" that he allowed her to share. (DW: The Masque of Mandragora) During the incident with the Sycorax on Christmas 2006, the translation circuit worked for all humans on the Sycorax ship, even though he was not, at the time, apparently aware of their presence. (DW: The Christmas Invasion) On yet another occasion, the Doctor told Elizabeth Klein that the telepathic field was limited to a certain radius around the TARDIS. (BFA: Survival of the Fittest)

It has been suggested on some occasions that the TARDIS translation also slightly modifies peoples' minds so that they do not consciously register that they should be unable to understand the languages they are 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 (DW: The Masque of Mandragora), while 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 Doctor reflected that the champagne the Brigadier had drunk was probably responsible for him asking the question (PDA: The Shadow in the Glass).

Companions
When the 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 Celtic. The Doctor then surmised that speaking Latin to a Latin speaker while using the translation circuit resulted in apparent Welsh. (DW: 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. (BFA: ...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. It made her perceive the Doctor as having a "stuffy Prussian accent" (BFA: Survival of the Fittest), while Vincent van Gogh, a native Dutch speaker, assumed that Amy was from Holland due to her having a similar accent to his own, both sounding Scottish. (DW: Vincent and the Doctor).

The Doctor
The Doctor's use of the circuit to aid his own communication was not well understood. A primary reason for the difficulty was the Doctor's apparent ability to speak many languages without the proximity of the TARDIS, such as when he was able to converse with Tritovores on San Helios, even though the TARDIS was on Earth. (DW: Planet of the Dead)

Another difficulty in assessing the Doctor's use of the circuit was the fact that his vast intelligence and longevity allowed for the possibility that he had simply learned some languages, and was able to speak them without using the circuit at all. When he used Madame de Pompadour's particular French accent to identify the century in which she lived (DW: The Girl in the Fireplace), it was difficult to tell whether he was using the circuit, or whether his stated love of France (DW: The Reign of Terror) had propelled him to study the development of its language closely enough to make such a determination on his own.

Furthermore, there were times when he was able to interject non-English words into his otherwise English speech — such as the phrases "nul points", "molto bene" and "Allons-y!" — without the circuit translating the phrase. The phrases were sometimes untranslated even when at least some of the recipients were not likely native speakers of English, in which the phrases were arguably loaned. (DW: The Parting of the Ways, Midnight, Voyage of the Damned, The End of Time)

Untranslated languages
Although the circuit could translate a vast array of languages in the universe- once even managing to translate when dealing with the language of the Vrill despite the fact that their language was based on smell rather than sound (BFA: Survival of the Fittest)-, it was occasionally stumped. If a language was one of which the Time Lords had no knowledge, as a language from the Dark Times present on Krop Tor once encountered by the Doctor and Rose Tyler, the translation circuit failed. (DW: The Impossible Planet) Some languages were too complex to be translated into human terms, such as the language of the Sittuun. (NSA: Night of the Humans)

The circuit may also have been unable to translate the native language of the Judoon, as the Doctor appeared to actually speak the language, rather than having it translated. (DW: The Stolen Earth)
 * This case is ambiguous, however, and many alternative explanations present themselves. The Doctor may have been observing a law of the Shadow Proclamation, trying to curry favour with the so-called "galactic police", or deliberately concealing his words from Donna. Doctor Who Confidential opined, perhaps jokingly, that the Judoon were "too thick" to have their language translated by the telepathic circuit.

Another language which was not translated by the TARDIS was Gallifreyan. While in a museum, the Doctor and Amy came across a homing box from a ship, which had Gallifreyan writing on it. The writing was not translated by the TARDIS, as proven when Amy asked what the writing said.
 * It is unknown why the TARDIS did not translate this, but it is possible that the TARDIS does not translate the Time Lord language, given that TARDISes were created by that race.

Whilst visiting Vincent van Gogh, the Doctor's TARDIS as covered in posters. These posters were in French and it is not known why it did not translate the words. (DW: Vincent and the Doctor)

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.