The Big Bang Two refers to the explosion caused by the Eleventh Doctor, "restarting" the universe after its destruction.
History
At "every point in history", all stars in the universe had gone supernova due to the unique circumstances of the TARDIS' destruction in 2010. The entire universe was destroyed and erased from existence except for the Earth and its Moon. The TARDIS' continuing explosion throughout all of history, putting itself in a time loop at the moment of its own destruction, acted as a replacement for the Sun, keeping humanity alive.
When the Pandorica was opened in 1996, it contained billions of atoms from the universe as it should have been. Some of these were the final traces of whole races un-made by the cracks in the universe. As "the perfect prison", the Pandorica was also designed to emit a restoration field to preserve or restore whatever was inside it, preventing prisoners from escaping even by dying. This combination of pre-collapse atoms and the restoration field had a restorative effect on a fossilised Dalek. This seemed impossible, because the Daleks had never existed, having been wiped from history; there should have been nothing to resurrect. The Eleventh Doctor thus reasoned that if the restoration field could be transmitted throughout all of space and time, the entirety of the former universe could be extrapolated from the atoms within the Pandorica.
The Doctor used a vortex manipulator to fly the Pandorica directly into the heart of the TARDIS' eternal explosion. Since the TARDIS was exploding at all points in history, the particles and restoration field were present at every point in time. A new universe was created based on the old one, but with the Doctor having never existed, as he had to 'seal' himself outside the universe in order to repair the cracks. "Restarting" the universe caused this new universe to proceed undamaged, because the TARDIS had not exploded. Thanks to the ability the crack in her room had given her to restore things erased by remembering them, in 2010, Amy Pond brought back, at least her parents and fiancé Rory, and at Rory's and her wedding, the Doctor after she focused on remembering him. (TV: The Big Bang)
Behind the scenes
- Because this new universe's survival hinged on certain omissions when compared to the history of the prior universe, it is unclear whether any particular previous Doctor Who adventures ever took place. However, it is assumed they did once Amy brought the Doctor back, as the content of River Song's diary were replaced. Also, in A Good Man Goes to War, the Thin One and the Fat One confirm that the Atraxi's thwarted attack on Earth and the Doctor's subsequent telling off still happened in the restarted universe. In the fourth series of The Sarah Jane Adventures, set after these events and featuring the Eleventh Doctor, confirms that all the events prior to The Big Bang had still happened in the restarted universe.