Meta-Crisis Doctor

The Meta-Crisis Tenth Doctor was created as a result of an instantaneous biological Meta Crisis. Regenerative energies present within the Doctor's severed hand combined with the human DNA of Donna Noble when she touched the hand's container. The force of the reaction shattered the container, allowing the energy to regenerate the entire missing body of the Doctor. This created a Time Lord/human hybrid, who possessed one heart, aged as humans did, and had no regenerative ability to avoid death.

Biography
As a result of growing partially from Donna's DNA, the new Doctor inherited parts of Donna's mannerisms. He maintained that he possessed 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. This motivated him to commit genocide against the Daleks, an act the original Doctor greatly condemned after the events of the Last Great Time War. The original Doctor described his clone as representing the way he had been during his ninth incarnation, prior to meeting Rose. Feeling his new, half-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 half-human self needed her, as he was angry and vengeful as he himself had been when he first met Rose. He told her that she had made him better, and now she had to do the same for his other, half-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 half-human Doctor offering to spend the rest of his life with her. Rose then asked both Doctors what the last thing they had said to her was when they were originally standing on Bad Wolf Bay (DW: Doomsday). The Doctor was unable to give her a direct answer: "Does it really need saying?" She posed the same question to the half-human Doctor, who whispered the answer in her ear.

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

The half-human Doctor would seemingly later return in an attempt to steal the Doctor's remaining incarnations, now sporting a goatee and black suit and calling himself the "Valeyard". However, this was later revealed to be an illusion created by Es'Cartrss of the Tactire. (IDW: The Forgotten)

Behind the scenes

 * Although he has the same appearance, memories, and basic personality of the Tenth Doctor, the new Doctor also exhibits several personality changes, in particular based upon that of Donna Noble. It is left deliberately ambiguous as to whether the new Doctor is actually a new incarnation of the original Doctor (if not strictly speaking a new regeneration) or a copy.
 * In the Doctor Who Confidential installment "The End of an Era", executive producer Julie Gardner confirms that the new Doctor does indeed say "I love you" when he whispers in Rose's ear.
 * On screen, this character is only ever referred to as the Doctor. To avoid confusion, fans have taken to referring to him by names such as the Clone Doctor, New Doctor, Meta-Crisis Doctor, Doctor Two, the human Doctor, Doctor 10.5, and Handy. None of these names are considered official, particularly references to him as the 11th Doctor, though David Tennant in Doctor Who Confidential acknowledges that this is expected to be a topic of much debate as time goes on.
 * The Doctor occasionally wore his blue suit even after the Meta-Crisis Doctor left the TARDIS with it, revealing that the Doctor either replaced it or owned more than one. (DW: Music of the Spheres, Dreamland, The Waters of Mars)
 * The clothes that the new Doctor chooses reflect the proposed outfit Tennant was to wear in the Series Three 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. (DW: Smith and Jones (commentary))
 * A number of fans have begun speculating as to whether or not this Meta-Crisis Tenth Doctor would eventually become the enigmatic figure known to the Sixth Doctor as the malevolent Valeyard. However, if the Meta-Crisis Doctor did succumb to his darker impulses, he still would not become the actual Time Lord Valeyard that the Sixth Doctor encountered, as the Meta-Crisis Doctor cannot regenerate into the Eleventh or Twelfth Doctors. This theory would seem to have gained some ground due to the events of the comic book mini-series, IDW: The Forgotten, in which the Meta-Crisis Doctor appears, now sporting a beard and calling himself the "Valeyard"; ultimately, however, this ended up being a form taken on by an insectoid creature that assumed his form inside the TARDIS matrix.

A "Pete's World" TARDIS
In the original script, the Doctor was meant to give TARDIS coral to the half-human Doctor under the assumption that a whole new TARDIS could be grown from it, but it was cut. This makes the notion strictly non-canonical. Nevertheless, the idea has popped up in several places: There is also the possibility that even if the coral grows to form a fully-developed TARDIS, it is unknown whether it would function since it was stated in DW: Rise of the Cybermen that the TARDIS draws its power from the universe and that Pete's World is the "wrong universe".
 * "The Doctor's Data" section of a "Doctor Who Adventure" magazine.
 * The Fact File for the episode on the official website, which stated that it was in the original script but later removed.
 * If one looks carefully at the 4th cover of the August 2008 Doctor Who Magazine, featuring Rose, Mickey, Jackie, and the Meta-Crisis 10th Doctor, it appears that, although covered up by actors' names, the duplicate Doctor seems to be holding something. This could possibly be a piece of TARDIS.
 * Inside the magazine, there is an excerpt of 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 complicatied the scene too much. However, the scene was filmed, and it was included in the Series 4 DVD boxset. The magazine went on to quote 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. The scene is included as a bonus feature with the Season 4 DVD box set released in November 2008.
 * Additionally, if you look closely, the Meta-Crisis Tenth Doctor is apparently holding the coral piece in the actual episode. The image is brief, however.