Telepathic circuits

The telepathic circuitry, (AUDIO: The Wrong Doctors) usually referred as telepathic circuits (TV: The Time Monster onwards) or telepathic induction circuits, (PROSE: State of Change, Shadowmind) was a TARDIS component 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) When Charlie Francis was turned into a telepath with telekinetic abilities after he was paralysed in an accident, he was able to use his abilities to communicate with Jo Grant by projecting himself into the Monk's TARDIS via its telepathic circuits. (AUDIO: The Rise of the New Humans) the Matrix of the Doctor's TARDIS, while possesing Idris used this function to lead Rory to an archived TARDIS control room, letting him drop the shields and allow the Junk TARDIS to land inside the archived room. (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. (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
Immediately after a psychic attack on the TARDIS, the Fifth Doctor used the telepathic circuits to pinpoint the source of the attack and find Adric, who had been kidnapped during the attack. Using the circuits took the Doctor, Nyssa and Tegan to Zaltys. (AUDIO: Zaltys)

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. (HOMEVID: 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 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, due to her focusing on the Doctor, 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 3W, an organisation led by. Indeed, the consciousness of Danny was trapped there, in the Nethersphere. (TV: Dark Water)

Clara Oswald managed to use on her own the telepathic circuits to drive the TARDIS to a specific fixed point in time. (COMIC: Four Doctors)

Grant used the telepathic circuits in conjuction with the Hazandra to help the Doctor find the other gemstones. (COMIC: Ghost Stories)

The Thirteenth Doctor used the telepathic circuits to focus on a watch that Yaz received from her grandmother. The TARDIS thus took the Doctor and her companions to the day before her grandmother received the watch during her first wedding. The telepathic circuit took the form of a knob, due to the Steampunk-ish controls of the console. (TV: Demons of the Punjab) The telepathic circuits could be extended from the console via a cord. (TV: The Battle of Ranskoor Av Kolos)

Other functions and effects
While recovering from regenerative collapse within his TARDIS, the Sixth Doctor used the TARDIS' telepathic circuits, combined with his own symbiotic nuclei, to construct a synaptic interface. (AUDIO: Piece of Mind)

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)

When the Eleventh Doctor became trapped outside time after he was forced to make a trip in a Type 1 TARDIS, he attempted to guide the ship back to reality by linking himself to its telepathic circuits when the conventional controls were unresponsive, but the TARDIS reacted by trying to unleash anti-time energies across the universe, only able to perceive the evil that the Doctor had witnessed in his travels despite his efforts to focus on other details. The first nine Doctors also became trapped in the rift as it created greater distortions in reality, but eventually the Eighth Doctor was able to escape and alert the Ninth, Tenth and Twelfth Doctors to what was happening. Once these three Doctors made contact with the Eleventh, they were able to help their other selves link their TARDISes' telepathic circuits to the Type 1's, convincing the Type 1 to end its efforts to destroy the universe. (COMIC: The Lost Dimension)

In the opening stages of the Last Great Time War, the Daleks seized control of a Battle TARDIS and used it to drain power from the Eye of Harmony. Unable to remotely take control of the ship via technological means, the Time Lords, on Susan Foreman's advice, sought the aid of the Sensorites who used their telepathic circuits to trigger the stolen TARDIS' self-destruct via its telepathic circuits. (AUDIO: Sphere of Influence)

References in invalid sources
When the TARDIS was crashing, Christel Dee mentioned that she 'really shouldn't have poked those telepathic circuits '. (WC: Doctor Who Minecraft)