The Titanic was a spaceship/cruise liner built by Max Capricorn Cruiseliners and in use during the early 21st century. Loosely based upon the design of the original Earth sea-going vessel RMS Titanic, this was primarily a tourist vessel, and it is unclear whether Max Capricorn, the owner, was aware of the name's significance. The ship had a humanoid crew, who were supplemented by a contingent of information and service robots called Hosts who provided assistance and advice to the passengers.
The vessel, under the command of Captain Hardaker, travelled to Earth in December 2008 in order for its passengers to experience the Earth holiday of Christmas. The vessel was bedecked with Christmas decorations, and its ship band performed Christmas songs, although some facts related to the holiday were skewed due to the ship's poorly-informed history expert, Bayldon Copper. The Doctor, fresh from leaving Martha Jones on Earth (DW: The Last of the Time Lords) and a brief encounter with his fifth incarnation (DW: Time Crash), accidentally collided his TARDIS with the Titanic, though neither vessel was damaged. The Doctor when boarded the vessel, effectively as a stowaway, and befriended several passengers as well as crewmembers Bayldon Copper and Astrid Peth.
Unknown to the passengers and crew, Capricorn had arranged for the Titanic to be destroyed in Earth orbit, in order to take revenge on the board of directors of his company, who had forced him out of his own firm. Captain Hardaker, terminally ill, had agreed to pilot the vessel in return for his family being taken care of. Capricorn, himself a cyborg, also boarded the vessel but expected to be protected from the disaster and later rescued.
Hardaker orchestrated matters so that the ship's shields would be lowered as three small asteroids approached the vessel. The resulting impact killed many of the crew and passengers (including Hardaker), but did not destroy the vessel outright. Capricorn had arranged for the Hosts' programming to be changed so they would begin killing any survivors. Over the next few hours, the Doctor and his friends, assisted by the Titanic's only surviving bridge crewmember, Midshipman Alonso Frame, worked to stabilize the ship's orbit, or at least stop it from crash-landing on Earth, which the Doctor feared would cause an explosion strong enough to wipe out all life on the planet.
Efforts to stabilize the orbit failed and Titanic began entering the Earth's atmosphere. After defeating Capricorn and rendering the Hosts no longer a threat, the Doctor was able to narrowly prevent Titanic from crashing into Buckingham Palace in London, and was able to restart the vessel's engines, returning it to orbit. Afterwards, the vessel, under the temporary command of Alonso Frame, was salvaged. (DW: Voyage of the Damned).
The four known survivors of the Titanic incident were: the Doctor (though he was never recorded as a passenger), Alonso Frame, Bayldon Copper (who departed the vessel to resettle on Earth), and businessman Rickston Slade. It's possible other crewmembers and passengers survived, if they were not killed by the Host.
Alternate timeline
In the alternative timeline orchestrated by the Trickster, the Doctor, having died a year earlier, was unable to prevent the Titanic from crashing to Earth in the heart of London. Although the resulting explosion proved the Doctor's prediction wrong in that it did not wipe out all life on Earth, it did decimate southern England, almost instantly transforming the United Kingdom into a third-world country and leading to a series of events that result in the ruined country becoming a police state. This timeline was negated when Donna Noble successfully restored the timeline, preventing the Doctor's premature death. (DW: Turn Left).