The Doctor is a Time Lord, an extraterrestrial from the planet Gallifrey, who travels in a time machine called the TARDIS (Time And Relative Dimension In Space) that allows him to reach any point in time and space and is larger on the inside than on the outside. The TARDIS originally had the ability to disguise itself according to its environment, but became "stuck" in the form of a police box after landing in London in 1963 and has remained in that shape ever since.
For the most part, and usually because the vessel's navigation system is old and unreliable, the Doctor explores the universe at random and uses his extensive knowledge of science and advanced technology to heroically avert the crises that he encounters. The Doctor has, at various times, been accompanied by companions who have chosen to travel with him for a variety of reasons.
The Doctor was considered a renegade by the Time Lords for his penchant of getting "involved" with the universe, in direct violation of official Time Lord policy. However, most of the time his actions were tolerated, especially when he has saved not just Gallifrey, but the universe, several times over. His standing in Time Lord society has waxed and waned over the years, from being a hunted man to even being appointed Lord President of the High Council (an office he did not assume for very long and eventually was removed from in his absence). In the end, though, he had always seemed quite content to remain a renegade and an exile. Ultimately, the Doctor found himself, by the time of his ninth incarnation, the last known surviving Time Lord.
Source: Wikipedia.org
After taking Rose to witness the destruction of the Earth 50 billion years in her future, the Doctor revealed to Rose that Gallifrey has been destroyed as a result losing a war, and that he is the last of the Time Lords.
In his ninth incarnation, the Doctor was by turns manic—possessed by a childlike sense of wonder and pure joy as he relished showing Rose the universe, past, present and future and experiencing it anew through her eyes—and somber, as he seemed deeply affected by the loss of his home and people. In a final battle with the Daleks on The Game Station in orbit above Earth in the year 201st Century, Rose Tyler absorbed the entrie time vortex, temporarily merging with the Heart of the TARDIS in roder to defeat the Daleks and save the Doctor. However, the vortex was killing her. To save her life, the Doctor absorbed the Vortex into his own body and used it to heal her. The net result was every cell in his body was damaged, forcing him to regenerate.
In his tenth incarnation, he's a man of action with a bit more hair, a bit slimmer, with a mole between his shoulderblades, is a wicked shot with a satsuma, has a right hand which is slightly newer than the rest of his new body... and he still cares deeply for Rose and wants her to remain his Companion.
When Rose met the Doctor, she was 19 and working as a shop assistant at Henrik's department store in Regent Street, London. She also had a boyfriend named Mickey Smith and lived in a council flat with her mother Jackie. Rose left school without taking her A-levels but did win the bronze medal in an under-sevens gymnastics competition at her junior school. Her father, Pete Tyler, died in 1987 in a car accident, the year after Rose was born.
One night after the shop closed she encountered mannequins coming to life in the basement of the building. The Autons were about to dispose of her when the Doctor saved her life, although he went on to destroy the building and her job in the process. She aided the Doctor in tracking down the hiding place of the Nestene Consciousness that was animating the Autons and subsequently helped defeat its plans of world conquest. She then joined the Doctor on his travels in the TARDIS.
In her travels with the Doctor, Rose has (among other things) seen the end of the world, encountered the Doctor's oldest enemy and learned about the consequences of tampering with history. The Doctor even modifed her Nokia 3200 mobile phone to be able to communicate across time and space, among other functions. She has nicknamed it the "Superphone".
During her first year with the Doctor, the words "Bad Wolf" followed the Doctor and Rose around, the phrase being scattered like clues through the places that they visited. In The Parting of the Ways, it was revealed that Rose was the Bad Wolf — the words were a message that she had left to herself in time and space when she absorbed the energies of the time vortex to save the Doctor and the Earth from the Daleks. The Doctor had just returned her home to place her out of harm's way, but "Bad Wolf" was a reminder that it was possible to get back to him. This led her to the point where she would absorb the energies, creating a predestination paradox and making it possible not just to destroy the Daleks but to leave those clues.
However, the energies she absorbed were destroying her body. The Doctor took those energies into himself, sacrificing his ninth incarnation and regenerating before Rose's eyes into the Tenth Doctor.
Source: Wikipedia.org
Rose is very bright, inquistive, and quick to grasp concepts no matter how strange or alien. She is quite capable of quick thought and action if the situation requires it, and is as likely to save the Doctor as to need saving. She shares his love of adventure and spirit of enthusiasm, and shows no hesitancy in going up agains the Doctor if they disagree.
While initially having difficulty coming to terms with the Doctor's regeneration, Rose seems to have accepted the Doctor's tenth incarnation as the same man she first met in Henrick's department store and considers her best friend in the universe. Having lost him once, she is fiercely protective of him.