Bad Wolf (entity)

After looking into the Heart of the TARDIS, the Time Vortex itself filled Rose Tyler, who then temporarily became the Bad Wolf entity. She scattered the two words throughout time as a sign to herself that she was linked to the Doctor, creating a looped ontological paradox.

Rose Tyler as the Bad Wolf
Newly empowered, she piloted the TARDIS back to the Ninth Doctor. A nimbus of light surrounded her and shone through her eyes. Rose's natural London accent vanished. She also possessed the destructive and creative powers of a goddess, enabling her to destroy an entire Dalek fleet and have an enormous influence over time and space.

While possessed of these gifts, the entity set out to spread the Bad Wolf meme through many worlds and times. In this way, the Ninth Doctor and Rose's past selves would eventually follow the clues that would enable them to defeat the Daleks, creating an ontological paradox in the process. The Bad Wolf destroyed the Dalek fleet and all the Daleks on Satellite Five, including the Dalek Emperor. She also altered reality to bring back Jack Harkness from death. (TV: The Parting of the Ways)

Jack later discovered he could apparently no longer die by any means, and it took him nearly a century and a half of subjective time before he finally learned that she was responsible. (TV: Utopia) As the instigator of Jack's immortality, Bad Wolf was therefore indirectly responsible for the event known as Miracle Day on Earth. (TV: The New World, et al)

Rose was unable to hold the power permanently without burning up, and the Doctor restored Rose's human self by absorbing the Vortex energy into his own body, a process that would prove destructive to his cellular structure and so prompt the Doctor's next regeneration. (TV: The Parting of the Ways) Rose subsequently lost most of her memory of her time as the Bad Wolf, saying that the knowledge had been locked, as if it was "forbidden", and it was stated that Rose trying to repeat it could cause the universe to explode. (TV: The Christmas Invasion)

Abilities
The Bad Wolf entity was incredibly powerful and could destroy an entire Dalek fleet by dividing all the atoms of the ships and the Daleks themselves. (TV: The Parting of the Ways) It also had the ability to resurrect already dead beings and make them immortal. (TV: The Parting of the Ways, Utopia) It could also form a force field to defend itself from Dalek weapons and spread messages, notably "Bad Wolf" to various points in time and space so that Rose and the Doctor would encounter the words which would eventually lead Rose to becoming the Bad Wolf in the first place. The Bad Wolf stated, "I created myself", which was a paradox. (TV: The Parting of the Ways)

Later references
Upon her meeting with the werewolf host, that entity noted that Rose also had "something of the wolf" in her. (TV: Tooth and Claw)

After being captured by Torchwood, the Doctor pretended Jackie Tyler was his companion Rose. He claimed that she had stared into the heart of the time vortex and aged fifty-seven years. She replied, "I'm forty!" (TV: Army of Ghosts)

While on the Valiant, commented to the Doctor how he "once had companions who could absorb the time vortex". (TV: Last of the Time Lords)

When Rose and the Doctor said good-bye, they met on a beach in Norway called Dårlig Ulv Stranden which Rose translated as "Bad Wolf Bay." (TV: Doomsday)

Rose whispered the words "Bad Wolf" to a dying Donna Noble in an alternate universe. After Donna remembered these words in the main universe, all the banners and signs in the Shan Shen marketplace read "BAD WOLF", including the sign on the TARDIS. (TV: Turn Left) This warned the Doctor and Donna of the impending danger of Davros and the Daleks. (TV: The Stolen Earth)

Also, all of Torchwood's records on file of the Doctor's companions were reportedly destroyed or corrupted in some way by a virus known as the "Bad Wolf virus". (TV: Love & Monsters)

When the consciousness of the Moment took form, it selected the form of Rose Tyler as the Bad Wolf entity. When the War Doctor commented that he could kiss her for giving him a way out of the dual-genocide he was prepared to commit, she remarked, "That's definitely gonna happen." The Tenth Doctor was visibly surprised to hear his past self mention the Bad Wolf, as he was unaware of the presence of Bad Wolf's likeness, and also because the War Doctor (as far as the Tenth and Eleventh Doctors knew) had never encountered Bad Wolf nor Rose Tyler. Due to the timelines rearranging themselves, the War Doctor (and his immediate successor) retained no memory of Bad Wolf. (TV: The Day of the Doctor)

Trivia
Bad Wolf (Entidade) Grand Méchant Loup
 * If the graffiti artist who wrote "BAD WOLF" on the TARDIS had written something else, the Bad Wolf would probably have used that name, as, in a way, it named itself after its own action of spreading its name across the universe. Every instance of BAD WOLF would also have been changed to match, including the name of the studio where Bad Wolf was "born".
 * This is similar to the naming of River Song, who, as the Eleventh Doctor said to her parents, "[You] named your daughter... after your daughter".