Telepathic circuits

The telepathic (induction) circuits were TARDIS components with multiple functions.

Communication
The telepathic circuits could be used to send messages telepathically to other people. The Third Doctor once used them to send the Time Lords a message about 's alliance with the Daleks when both sides tried to trigger a war between Earth and Draconia. (TV: Frontier in Space, Planet of the Daleks) Idris used this function to lead Rory to an archived TARDIS control room, letting him drop the shields and allow the Junk TARDIS to enter the Doctor's TARDIS. (TV: The Doctor's Wife) It was also somehow connected to the Doctor's use of the TARDIS phone, and could be accessed externally by powerful beings such as the Beast. (COMIC: The Beast Is Back In Town)

Translation
Additionally, this element of the TARDIS systems allowed its occupants to understand and speak languages other than their own. The TARDIS' pilot seemed to be an integral part of the telepathic circuits; if the Time Lord were unconscious or severely injured, the translation effect might not work correctly. (TV: The Christmas Invasion)

While the Doctor's companions heard and spoke their native languages and did not perceive the translation effect, the Doctor himself was aware of the translation and whatever local language was being spoken. (TV: The Fires of Pompeii, The Girl in the Fireplace)

Should someone benefiting from the translation effect attempt to override it and deliberately speak the local language, it came across to the locals as garbled. It also seemed the listener (at least on Earth) heard something similar to the language of the speaker's native culture in that time period. It is possible that the listener heard the native language of the speaker without comprehending it. (TV: The Fires of Pompeii)

The translation circuits translated writing as well as speech. (TV: The Fires of Pompeii, A Good Man Goes to War) They did not work on languages that the Time Lords themselves don't know, suggesting it acted like a translator for a vast number of languages. (TV: Four to Doomsday, The Impossible Planet) The translation circuits could translate all text in an area to show one specific message, as when the TARDIS translated all text on Shan Shen to "Bad Wolf". (TV: Turn Left)

Navigation
The Eleventh Doctor used the circuits to pinpoint the saddest memory in Amy's life. Even if the memory seemed too ridiculous to be her saddest memory, Amy was encouraged by the Doctor to replace the ice cream she lost in 1994 to make her past self happy. (TV: Good Night) The Eleventh Doctor linked Clara Oswald to the circuitry in order to retrieve from her mind the visual coordinates of Trenzalore. (TV: The Name of the Doctor)

The Twelfth Doctor, trying to find out what the creature that he believed followed him was, hooked Clara up to the telepathic circuits, taking the form of a gel-like substance she could put her hands in and used them to extrapolate her timeline from birth to death. By focusing on a moment in her past, she could then effectively control the TARDIS to visit it. As Clara got distracted by a phone call from Danny Pink, she inadvertently took them into his past instead of hers. Later, the Doctor used a trace of Clara left in the circuits to try again while Clara wasn't there and met Orson Pink, someone from about a century into Clara's future trapped at the end of the universe. When the Doctor was knocked out and they seemed to be under threat, Clara tried to use the telepathic circuits to fly them to safety once more. Clara succeeded in flying the TARDIS in this manner, but it took them far into the Doctor's past to when the First Doctor was a boy. (TV: Listen)

After the loss of Danny Pink, the Twelfth Doctor invited Clara to link again to the circuits, question about the location of her lover, in order to retrieve him. They were brought to the headquarters of the 3W, an organisation led by Missy who claimed to manage afterlife. Indeed, the consciousness of Pink was trapped there, in the Nethersphere. (TV: Dark Water)

Other functions
In an alternate future, the circuits of a dying Doctor's TARDIS awakened in Clara Oswald's mind the memories from an aborted timeline. (TV: The Name of the Doctor)

The telepathic circuits could also be used to alter minds in other ways. The Seventh Doctor used his TARDIS' telepathic circuits to erase some of his useless memories, accidentally erasing Ace's memory as well. Soon after, the Doctor used the circuits to bring forth the personality of his third incarnation when he had to disarm a cobalt bomb and was uncertain of his present self's expertise in handling a delicate piece of machinery. (PROSE: Timewyrm: Genesys) When the Seventh Doctor restored Ace's memories to her, he accidentally gave her some of Melanie Bush's also, allowing her to remember Paradise Towers. (PROSE: Timewyrm: Revelation) On another occasion, the Seventh Doctor transferred his entire conscious mind into the TARDIS telepathic circuits, erasing his own memories while protecting his mind from a telepathic assault that was trying to provoke him into killing Ace. This left him an amnesiac until he made contact with the circuit again, restoring his memory while using the circuit as a "barrier" to prevent the Valeyard from attacking him again. (PROSE: Matrix)