Tenth Doctor's hand

The right hand of the Tenth Doctor was severed by the Sycorax leader shortly after the Doctor's regeneration. At some point, Captain Jack Harkness had acquired it and brought it to the Torchwood Three hub.

The hand was kept well-preserved on its own and remained alive and somewhat responsive despite getting separated from its original body. It became an organism independent from the Doctor which could flex itself under its own power, and eventually, part of a new gestalt Doctor following a meta-crisis.

History
After the Ninth Doctor regenerated into his tenth incarnation, the hand was attached to his arm. 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. (TV: Children in Need Special)

During a sword fight with the Sycorax leader on the deck of their ship, the leader cut the Doctor's right hand off and it fell somewhere in London. Because he was within the first fifteen hours of his regeneration, he regrew the hand. (TV: The Christmas Invasion) The excess regeneration energy was siphoned off of the Third Doctor. (PROSE: The Christmas Inversion)

After Captain Jack Harkness travelled from 200,100 to 1869, he lived through the 20th century and eventually found the hand. He 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. (TV: Utopia) 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. (TV: Day One)

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

When the Doctor arrived at Cardiff Bay, the hand reacted, glowing and moving. (TV: End of Days) 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. (TV: Utopia)

The Doctor fixed Jack's Vortex Manipulator so they could travel to 2008. When they encountered the Master, he revealed that he had put Richard Lazarus' genetic technology into his laser screwdriver. Using the Doctor's biological code gained from the hand, the Master aged the Doctor a hundred years. (TV: The Sound of Drums) The Year That Never Was followed, after which the Doctor reclaimed his severed hand and kept it aboard his TARDIS in the control room. (TV: 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 imminent birth. (TV: The Poison Sky, The Doctor's Daughter)

The Doctor's hand remained in his TARDIS and was present for the Doctor's regeneration when he was shot by a Dalek. (TV: The Stolen Earth) After the Doctor had healed the damage caused by the shot, however, he channeled the excess regeneration energy that would have changed his appearance into a bio-matching receptacle, his hand, leaving him healed while remaining in his tenth incarnation, (TV: Journey's End) but still using up a regeneration in his cycle. (TV: The Time of the Doctor) Donna 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 Doctor mixed with Donna's persona and attitude. (TV: Journey's End)

Behind the scenes
David Tennant provided the cast for the hand during the filming of Utopia. This cast was retained during the remainder of the prop's time on the show.

In Jack's flashback to 1999 in TV: Fragments, the Doctor's hand in the jar can be seen in the the Hub despite the scene taking place years before it would be cut off in 2006. No narrative explanation has been given for this discrepancy.