The Doctor's hand was the severed right hand of the Doctor which he lost shortly after he regenerated into his tenth incarnation.
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During a sword fight with the Sycorax leader on the deck of their ship, the leader slashed his right hand off and it fell somewhere in London. Because he was within the first fifteen hours since his regeneration, he was able to regrow the hand. (DW: The Christmas Invasion)
Captain Jack Harkness traveled from 200,100 to 1869 using a Vortex Manipulator. After living through the entire 20th century, he 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)
Jack preserved the hand in a tank or jar which was destroyed by Carys Fletcher. (TW: Day One) However, the container was later seen to be intact, presumably repaired or replaced by Jack.
After the Doctor came to refuel at the Rift, Captain Jack joined the Doctor and brought the severed hand with him. After traveling 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)
Just as the Doctor and Donna Noble were saying goodbye to Martha after the Sontaran invasion of Earth, the TARDIS set off by itself, causing Donna to briefly fall down beside the hand and have Martha explain its origins to her. (DW: The Poison Sky) The TARDIS had brought the trio to the planet Messaline, having detected that the Doctor had a daughter Jenny.
Jenny was created when a sample of The Doctor's DNA was put in a Progenation Machine, thus creating Jenny and attracting the TARDIS, creating a paradox. (DW: 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 channeled the excess regenerative energy that would have changed his appearance into a bio-matching recepticle in the form of his hand, thus leaving him healed while remaining in his tenth incarnation.
However Donna eventually touched the energised hand which caused a Meta-Crisis from which a new version of the Doctor grew. The new Meta-Crisis Doctor had the intelligence of the original Doctor mixed with Donna's persona and attitude. Later Donna was hit by an electrical blast from Davros which triggered a second Meta-Crisis within Donna.
This gave her the same intelligence that the Doctor had, thus technically making two of the same person; Meta-Crisis Doctor and Donna. Unfortunately the Time Lord knowledge that now resided within Donnas' brain was too much and almost killed her until the Doctor wiped her memory. (DW: Journey's End)
Behind the scenes
- David Tennant's hand did not provide the cast for the hand.
- 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.