Tenth Doctor's hand

The Doctor's hand was the severed right hand of the Doctor which he lost shortly after his regeneration into his tenth incarnation.

History
After the Ninth Doctor regenerated into his tenth incarnation, the hand was attached to his body. The newly-regenerated Doctor noted a "slight weakness in the dorsal tubercle" around his hand, but this was not mentioned again during the remainder of the time this hand was attached to his body. (DW: Children in Need Special)

During a sword fight with the Sycorax leader on the deck of their ship, the leader slashed the Doctor's right hand off and it fell somewhere in London. Because he was within the first fifteen hours of his regeneration, he was able to regrow the hand. (DW: The Christmas Invasion) After Captain Jack Harkness travelled from 200,100 to 1869, he eventually found the hand and used it to detect the Doctor whenever he came to the Cardiff Rift to refuel, hoping to find a version of the Doctor which coincided with his own timeline. (DW: Utopia) The precise details of how it detected the presence of the Doctor are unknown. Jack kept the hand preserved in a jar which he kept on his desk. Despite being severed, it did not decay and continued to move and twitch. (TW: Day One)

When Carys Fletcher escaped, she broke the jar, distracting Jack enough for her to escape. (TW: Day One)


 * However, the container was later seen to be intact, presumably repaired or replaced by Jack.

When the Doctor arrived at Cardiff Bay, the hand started to react, glowing and moving. (TW: End of Days) After the Doctor came to refuel at the Rift, Captain Jack joined the Doctor and brought the severed hand with him. After travelling to the year 100,000,000,000,000, the Doctor, Martha Jones, and Jack met Professor Yana, who then became the Master. He took the Doctor's hand from Jack and escaped in the Doctor's TARDIS after he regenerated. (DW: Utopia)

The Doctor was able to fix Jack's Vortex Manipulator so they could travel to 2008. When they encountered the Master, he revealed that he put Richard Lazarus' genetic technology into his laser screwdriver. By using the Doctor's biological code he gained from the hand, the Master was able to age the Doctor 100 years. (DW: The Sound of Drums) What followed were the events of The Year That Never Was, after which the Doctor reclaimed his severed hand and kept it aboard his TARDIS in the control room. (DW: Last of the Time Lords)

After the Sontaran invasion of Earth, the TARDIS set off by itself to Messaline, with the hand apparently reacting to Jenny's later birth. (DW: The Poison Sky, The Doctor's Daughter)

The Doctor's hand remained in his TARDIS and was present for the Doctor's apparent regeneration following his being shot by a Dalek. (DW: The Stolen Earth) After the Doctor had healed from the damage caused by the shot, he channelled the excess regenerative energy that would have changed his appearance into a bio-matching receptacle in the form of his hand, thus leaving him healed while remaining in his tenth incarnation. (DW: Journey's End)

However, Donna eventually touched the energised hand, which caused a new version of the Doctor to be generated. The new Meta-Crisis Doctor had the intelligence of the original Doctor mixed with Donna's persona and attitude. (DW: Journey's End)

Behind the scenes

 * David Tennant did not provide the cast for the hand during it's time on Torchwood, however he DID provide the cast during the filming of Utopia. This cast was retained during the remainder of the props time on the show until Journey's End.
 * Although Journey's End doesn't go into detail, it is possible to speculate based upon the Doctor's statements to Donna, Rose and Jack that he kept the preserved hand in the TARDIS console room (rather than putting it into storage) for very likely the purpose shown in Journey's End as a vessel to stave off a regeneration. Speculatively, had the hand not formed the Meta-Crisis Tenth Doctor or actor constraints not forbidden it, the Tenth Doctor may've been able to use the hand to avoid regeneration indefinetely and, arguably, lived forever.