Meta-Crisis Doctor

The Meta-Crisis Doctor was a Time Lord/human hybrid and a human incarnation of the Tenth Doctor, who possessed one heart, aged as humans did and had no regenerative ability to avoid death. He was created by an instantaneous biological meta-crisis and was the final result of the Tenth Doctor's aborted regeneration.

However, this Doctor was born in a separate body than the original Doctor, existing as a separate person with DNA sampled from Donna Noble. After destroying the Daleks, this Doctor was left behind with Rose Tyler in Pete's World.

Pre-creation
Mawdryn attempted to force the Fifth Doctor to use up his eight remaining regenerations to end his follower's cycle of perpetual rebirth, but this was rendered unnecessary when Alistair Gordon Lethbridge-Stewart made physical contact with his younger self and a discharge of temporal energy was released that allowed Mawdryn and his followers to die. (TV: Mawdryn Undead)

After the TARDIS became "stalled in the equivalent of a galactic lay-by", the Sixth Doctor had a worried thought of Peri Brown growing old and dying in the TARDIS, while he would "go on regenerating until all [his] lives [were] spent." (TV: Vengeance on Varos)

When exposed the Valeyard's alliance with High Council to the Sixth Doctor at his trial, he revealed that the Valeyard was acting as the prosecutor for the trial in exchange for the Doctor's remaining regenerations. (TV: The Ultimate Foe)

When Ace was sent into the Seventh Doctor's mind, she discovered a room with thirteen cubicles, seven of them empty, while the other six contained shadowy white figures, representing the Doctor's future incarnations. (PROSE: Timewyrm: Revelation)

After using a Deathworm Morphant to posses a human body, tried to use the Eye of Harmony to steal the Eighth Doctor's remaining regenerations to heal himself, but his plans were foiled when Grace Holloway sent the TARDIS' into a temporal orbit. (TV: Doctor Who)

Origin
On Christmas 2005, the newly regenerated Tenth Doctor lost his hand in a sword fight against the Sycorax leader. (TV: The Christmas Invasion) Jack Harkness later retrieved the hand and returned it to the Doctor in a jar in the year 100 trillion. (TV: Utopia) The Doctor's hand remained in the TARDIS until the Doctor was grazed by the blast from a Dalek gunstick, triggering the Time Lord's eleventh regeneration. After healing the damage to his body with the regeneration energy, he channelled the excess energy into his hand, a bio-matching receptacle, allowing him to heal but not change his appearance and personality. (TV: Journey's End) Even though the regeneration was aborted before it could be completed, the fact that the Doctor syphoned off the energy that would've changed his appearance after healing his injuries, he used up the same amount of regenerative energy as a full regeneration. This, as the Eleventh Doctor later told Clara Oswald, meant that his previous incarnation used up two regenerations due to "vanity issues." (TV: The Time of the Doctor)

Shortly after this aborted regeneration, the Daleks attempted to destroy the Doctor's TARDIS. Still inside the TARDIS during this apparent destruction, Donna Noble touched the energised hand's container. The regenerative energies present within the hand combined with Donna's human DNA, causing an instantaneous biological Meta-Crisis. The force of the reaction shattered the container, allowing the energy to regenerate the entire missing body of the Doctor. In this process, the "Meta-Crisis Tenth Doctor" was created. He immediately dematerialised the TARDIS, saving it and Donna.

After putting on a blue suit, the Doctor and Donna left the TARDIS and tried to attack Davros and the Daleks with his new weapon. Both were electrocuted by Davros. After the now-part-Time Lord Donna defeated the Daleks, she, the Meta-Crisis Doctor and the original Doctor teamed up to return all but one of the planets to their rightful places. The Meta-Crisis Doctor was then prompted by Dalek Caan to fulfil a prophecy by destroying the Daleks. He chose to do so and ignored pleas from both Davros and Donna. He proceeded to overload the Dalekanium power feeds, causing virtually the entire New Dalek Empire to explode, from individual Daleks to the Crucible itself. (TV: Journey's End)

Exile in Pete's world
Feeling his new, part-human counterpart was too dangerous to be left to his own devices, the Doctor entrusted him to Rose Tyler, taking the two of them back to Bad Wolf Bay in Pete's World. The Doctor told Rose that his part-human self-needed her, that he was angry and vengeful as he himself had been when he first met Rose. He told her she had made him better, and now she had to do the same for his other, part-human self. Rose objected to this at first, insisting that the new Doctor was not really him, despite the Doctor's assurance that they were the same man and the part-human Doctor offering to spend the rest of his life with her. Rose asked both Doctors what the last thing they had said to her was when they were originally standing on Bad Wolf Bay. The Doctor was unable to give her a direct answer: "Does it need saying?" She posed the same question to the part-human Doctor, who whispered it in her ear.

Seemingly overcome with the revelation of his answer, Rose grabbed the lapels of the part-human Doctor and pulled him into a kiss, which he reciprocated. The Doctor looked on sadly yet stoically for a moment before returning with Donna to the TARDIS and leaving the parallel world. As the doors of the TARDIS slammed shut, Rose and the part-human Doctor broke from their kiss. The part-human Doctor took Rose's hand and watched the TARDIS dematerialise. (TV: Journey's End)

Personality
As a result of growing partially from Donna's DNA, the new Doctor inherited some of Donna's mannerisms. He claimed he had the same memories, thoughts and feelings of the original Doctor up to the point of his aborted regeneration, making them essentially the same man with physiological differences. However, the original Doctor was quick to point out that his new double was "born in battle, full of blood, anger and revenge". In this way, the Tenth Doctor described his duplicate as representing the way he had been during his former incarnation before meeting Rose. This motivated him to commit attempted genocide against the Daleks, an act the original Doctor condemned after the events of the Last Great Time War. Regardless of their differences, he still firmly declared he was the Doctor, seeing he had a right to the title even if Donna Noble wasn't so certain. (TV: Journey's End)

Behind the scenes

 * Although he had the same appearance, memories, and basic personality of the Tenth Doctor, the new Doctor also exhibited several personality changes, in particular, based on Donna Noble.
 * In the Doctor Who Confidential episode End of an Era, executive producer Julie Gardner confirmed that the intention was that the new Doctor did indeed say "I love you" when he whispered in Rose's ear.
 * On screen, this character was only ever referred to as the Doctor. To avoid confusion, fans took to referring to him by names such as the New Doctor, Meta-Crisis Doctor, Ten II, Tentoo, the human Doctor, Doctor 10.5, Tenth Doctor Duplicate, Doctor/Donna, John Noble and Handy. None of these names are considered official, particularly references to him as the 11th Doctor. In the shooting script for the episode, he is simply named as "THE DOCTOR #2". David Tennant in Doctor Who Confidential acknowledged that the name of this particular incarnation of the character was expected to be a topic of much debate as time went on.
 * As recently as March 28th, 2018, official social media accounts for Doctor Who have referred to this incarnation by the fandom name "Tentoo".
 * The Doctor occasionally wore his blue suit even after the Meta-Crisis Doctor left the TARDIS with it, indicating that the Doctor either replaced it or owned more than one. (TV: Music of the Spheres, Dreamland, The Waters of Mars)
 * The clothes that the new Doctor chose reflected the proposed outfit Tennant was to wear in the series 3 opener, Smith and Jones, with the Doctor wearing a red t-shirt under his blue suit as opposed to the light blue shirt/tie combo that was eventually used. (DCOM: Smith and Jones)
 * His fashion sense is also reminiscent of the Ninth Doctor's, consisting of matching coloured jacket and trousers with a casual collarless shirt underneath, in contrast to all other incarnations of the Doctor wearing a dress shirt. (TV: Rose etc Al.)
 * A number of fans have speculated whether this Meta-Crisis Tenth Doctor would eventually become the Valeyard. However, among other things, dialogue in The Name of the Doctor stated "Valeyard" as a future name for the Doctor that was yet to come. The idea of the Meta-Crisis Doctor becoming the Valeyard was referenced in the comic story The Forgotten, in which Es'Cartrss took the form of a bearded Tenth Doctor claiming to be both the Meta-Crisis Doctor and the Valeyard.
 * There was much fan speculation as to whether or not the Doctor had used up one of his regenerations when he channelled the excess regeneration energy into his hand. In the 2013 Christmas special, The Time of the Doctor, the Eleventh Doctor reflected on his regeneration cycle and confirmed that these actions did indeed use up a regeneration. He dismissed the Tenth Doctor's reluctance to change his face as "vanity issues".
 * The concept that a Doctor would grow from the Doctor's severed hand who would end up with Rose was planned since The Christmas Invasion. Russell T Davies states on the commentary track for Journey's End that he had the idea in mind that just prior to the Tenth Doctor's regeneration, a scene would depict him growing a clone of himself from his severed hand, and send him off to live his life with Rose. The concept was ultimately brought forward, and was developed into its own entire story arch, due to the "timing being perfect".

A "Pete's World" TARDIS
In the original script, the Doctor was meant to give TARDIS coral to the part-human Doctor on the assumption that a whole new TARDIS could be grown from it, but it was cut as stated earlier.

Nevertheless, the idea has popped up in several places:
 * "The Doctor's Data" section of a Doctor Who Adventures magazine.
 * The Fact File for the episode on the official website, which stated that it was in the original script but later removed.
 * Inside the magazine, there is an excerpt of the script with a statement that this part of the scene made it all the way to the last cut, but the producers decided that it just complicated the scene too much. However, the scene was filmed and it was included in the series 4 DVD boxset. The magazine quoted Russell T Davies' opinion that the Meta-Crisis Tenth Doctor and Rose Tyler have a piece of TARDIS coral, which they can use to grow their own TARDIS.
 * In Russell T Davies' book The Writer's Tale, the full original draft of the Bad Wolf Bay script has the Doctor stating that it takes "thousands of years" to grow a TARDIS. However, Donna uses her newfound Time Lord knowledge enhanced by human intuition to overcome this problem by suggesting: "...if you shatterfry the plasmic shell and modify the dimensional stabiliser to a foldback harmonic of 36.3, you accelerate growth by the power of 59!" Which would presumably enable a TARDIS to grow within a human lifetime.