The Doctor's TARDIS key

The Doctor possessed a TARDIS key which they used throughout their lives for various functions, primarily unlocking the TARDIS. Sometimes, they also possessed a spare key. According to the Tenth Doctor, the key maintained a symbiotic link to the TARDIS that could be used to pinpoint the craft within a hundred years. (COMIC: The Forgotten)

History
When on Destination, blackmailed the First Doctor for his TARDIS key, stealing the TARDIS and abducting Ian Chesterton and Barbara Wright. After the schoolteachers had overpowered the Master and returned to Destination, they presented the key back to a jubilant Doctor. (AUDIO: The Destination Wars)

The Doctor sometimes also left their own key with one of their companions. On one occasion when the Third Doctor thought he was soon to die, he gave his TARDIS key to Jo Grant, so it wouldn't be lost along with him. (AUDIO: The Tyrants of Logic)

In some cases, the Doctor simply needed their companion to retrieve the TARDIS and get them all out of danger, such as when the Seventh Doctor temporarily gave his key to Roz Forrester. (AUDIO: The Jabari Countdown)

The Eighth Doctor once left his with Tamsin Drew as a statement of trust, in the hopes that she would leave the Monk and rejoin his side of the fight. (AUDIO: Lucie Miller)

After the War Master had been fatally wounded by Chantho, he made to steal the Doctor's TARDIS. The Tenth Doctor attempted to use his key to enter the ship only for the Master to deadlock the ship's doors. (TV: Utopia) When the Tenth Doctor regenerated, (TV: The End of Time) he kept the key on his person. Seeking shelter from the Atraxi, the Eleventh Doctor tried to use it to enter the TARDIS but his ship barred him access until she repaired herself, the key glowing when it was done. (TV: The Eleventh Hour)

In his many meetings with the Doctor, Winston Churchill often made attempts to steal the Doctor's TARDIS key. In the aftermath of the Ironside Incident, Churchill succeeded but Amy Pond noticed the theft and took it back. (TV: Victory of the Daleks)

At the onset of the Siege of Trenzalore, the Eleventh Doctor surrendered one of his TARDIS keys to Tasha Lem when she sent him and Clara Oswald down to Trenzalore to investigate the mysterious message. Having since shaved his head out of boredom, the bald Doctor had concealed another key in his wig, using it to summon the TARDIS around him and Clara, saving them from a nest of Weeping Angels. (TV: The Time of the Doctor)

The Twelfth Doctor kept several spare keys scattered in the TARDIS control room. (TV: Dark Water)

After the Twelfth Doctor regenerated, the TARDIS console room exploded due to the damage it took from the intensity of the regeneration energy. The newly born Thirteenth Doctor was thrown out of the TARDIS from the explosion, causing her to free fall to the Earth's surface. (TV: Twice Upon a Time) As she fell, she lost her TARDIS key along with everything else in her pockets. (TV: The Woman Who Fell to Earth) When she reunited with the TARDIS, she explained the loss of her key, causing the ship to open its doors of its own volition. (TV: The Ghost Monument)

Function
The barrier on the Doctor's TARDIS, a Type 40, was a double curtain trimonic. As such, the Chancellery Guard needed the model's cypher indent key to gain entry without the actual key, as opposed to other methods of forced entry. (TV: The Deadly Assassin) In its police box disguise, entry was allowed by inserting a key into the lock, just like a real police box. (TV: Doctor Who, Rose and others) However, the locking mechanism did not respond to police-issued keys. (TV: Blink) According to Susan Foreman, the lock had a defence mechanism in which there were twenty one different holes inside the lock, one being the right place and other twenty being wrong. If a mistake was made, the whole inside of the lock would melt. (TV: "The Survivors") The Brigadier was prevented from opening the TARDIS in the Third Doctor's absence, because of the metabolism detector on the lock. (TV: Spearhead from Space) Nevertheless, there were occasions on which strangers were able to successfully use the Doctor's key, suggesting either a flaw in the metabolism detector or a choice on the part of the TARDIS. (TV: The War Machines, Doctor Who, Blink) The lock could be manually secured from inside the TARDIS, preventing even authorised individuals from using the key to unlock the doors from the exterior. (TV: The Dalek Invasion of Earth, Utopia) At some point shortly before the Tenth Doctor's regeneration, the TARDIS had a system that allowed him to lock the TARDIS remotely using a fob (as a joke, the TARDIS roof light flashed and a alarm chirp sound was heard, similar to that used on vehicles on Earth). He was also able to open the door remotely. (TV: The End of Time)

There were occasions on which the Doctor or their companions needed to use the key to lock the doors, (TV: "Strangers in Space") while on others (TV: Spearhead from Space, The Christmas Invasion) the act of merely closing the doors locked the TARDIS. It was evident that the TARDIS' lock could be set to secure either automatically or manually.

However, a key was not always needed to open a TARDIS. During the Doctor's first encounter with River Song (from his point of view), she remarked that she had witnessed his future self open the doors of the TARDIS by snapping his fingers. The Tenth Doctor reacted with disbelief, but later successfully opened the doors in this fashion. (TV: Forest of the Dead) After regenerating into his eleventh self, the Doctor again opened the TARDIS by snapping his fingers, demonstrating that this method had become a reliable way for him to do so. (TV: The Eleventh Hour) The Doctor did, however, continue to make regular use of his key to unlock the TARDIS. (TV: The Time of Angels, Flesh and Stone, et. al.) After losing the key in their thirteenth incarnation, the Doctor simply told the TARDIS what had happened and she opened up on her own. (TV: The Ghost Monument)

Design
The key to the Doctor's TARDIS took several forms, as the key was able to change shape with the chameleon circuit. (AUDIO: The Alchemists)

At some point in his third life, the Doctor began experimenting with new designs for his key, coming up with several alternative designs before settling on a spade-shaped silver key. After his regeneration, the Fourth Doctor experimented with even more radical designs, which included a double-helix apparently made of plastic, before eventually going back to the spade and Yale styles. (PROSE: Interference - Book Two) The Yale key was also used by the Fifth Doctor. (TV: Four to Doomsday)

The Seventh Doctor used yet another design, incorporating the Seal of Rassilon with a fan-shaped motif. (TV: Ghost Light) The Seventh Doctor eventually went back to the spade key, which the Eighth Doctor initially continued to use. (TV: Doctor Who) At some point, the Eighth Doctor returned to using the Yale key. (PROSE: The Crooked World, History 101)

The Ninth, (TV: Aliens of London, Father's Day) Tenth, (TV: 42) and Eleventh Doctors (TV: The Eleventh Hour) used Yale-type keys in a gold/brass colour as opposed to earlier Doctors' silver-coloured Yale keys.

Abilities
Beyond simply unlocking the TARDIS doors, the Doctor's TARDIS key has often displayed other uses and characteristics.

The Doctor's TARDIS key could be used to open TARDISes other than his own. (TV: The Mark of the Rani)

The TARDIS key glowed and grew warm when the TARDIS was about to materialise nearby (TV: Aliens of London) or when an incapacitated TARDIS had finished rebuilding itself. (TV: The Eleventh Hour)

When the TARDIS fell through a wound in time, the key was hot and glowed, indicating that it was still connected to the TARDIS, leading the Ninth Doctor to use the key in order to call the TARDIS. The Doctor warned that if the key was touched by anything, it would cause the now-re-materialising TARDIS to disappear. After Rose Tyler came into contact with her younger self, it caused a Reaper to emerge which consumed the Doctor. As it tried to fly away, the Reaper touched the key which destroyed it and caused the TARDIS to disappear. (TV: Father's Day)

As a part of the TARDIS, keys had low-level perception filter properties. The Tenth Doctor was able to tune the keys into the psychic network created by the Master to extend these properties to an individual wearing a key. (TV: The Sound of Drums)

The key could remotely lock the TARDIS from a distance, similar to some car keys. (TV: The End of Time)

The key could be used to place the TARDIS one second out of sync with the rest of the universe, preventing it from being seen. (TV: The End of Time)

The key could be modified to track and locate the TARDIS, allowing the Doctor to find the TARDIS if it was within a hundred years of his position. (COMIC: The Forgotten)

When Sarah Jane Smith left the key in the lock without opening the door, it cancelled the pause control. The TARDIS travelled to its set coordinates on Earth, instead of remaining in the copy of Devesham created by the Kraal. (TV: The Android Invasion)

The Eleventh (TV: The Time of the Doctor) and Twelfth Doctor (TV: Death in Heaven) were able to summon the TARDIS to him using the key.

The keys were very resilient to damage, to the point that Clara Oswald believed they could only be destroyed by being dropped into lava. (TV: Dark Water)

Information from invalid sources
Shortly after regenerating, the Seventh Doctor found that the TARDIS' proto-anodizing discorporators had short-circuited the molecular quark overload. After attempting to use his TARDIS key, he declared it was "no good", and that he'd have to use the dimorphic inertia system. (NOTVALID: Untitled)

Other matters

 * The original spade-type TARDIS key was designed by Jon Pertwee and began use in Season 11.
 * When Pertwee left, he kept the original prop. Two new slightly different keys were made for Tom Baker's first season.
 * The Rassilon-style key was designed by Mike Tucker.
 * The prop used for the TV movie was a BBC-licenced replica of the classic spade key, sold at the time by now-defunct American memorabilia catalogue 800-TREKKER.
 * The 2005 key was a standard nightlatch key made by UK lock manufacturer ERA, and it was the real key to the working lock on the TARDIS prop. Spares were stored in a secret compartment inside the TARDIS prop.