The Big Bang is the 13th and final episode of Series 5 of Doctor Who. It stars Matt Smith as The Eleventh Doctor, Karen Gillam as Amy Pond and Caitlin Blackwood as a younger version of herself, Arthur Darvill as Rory Williams and Alex Kingston as River Song. It is the conclusion to the events of the last episode, The Pandorica Opens.
At the conclusion of the last episode, The Doctor had been imprisoned in The Pandorica, the universe's most perfect prison. Rory Williams, who had seemingly returned to life, turned out to be a plastic replica, who then shot Amy. Also, the entire universe is on the brink of destruction. It seems like a dire, unsolvable situation.
This episode begins with a young Amelia Pond listening to her mother talking about how she hopes that her daughter hasn't fallen under the sway of star cultists...those who imagine that the night sky should have something called "stars". Amelia seems to remember something else, and her belief is justified when she gets a strange invitation to go The British Museum: where she finds the Pandorica. Opening the Pandorica, she finds its occupant, an older version of herself, who calmy states:
"Okay kids, this is where it gets complicated."
And thus starts an adventure that manages to solve all the conundrums brought up in the past 12 episodes, as well as saving the Doctor and rebooting the universe from scratch. To do this, The Doctor locks himself in the Pandorica, a perfect prison, which can therefore preserve information about the universe even as it is destroyed. All The Doctor has to do is lock himself in the Pandorica and use it to go back to the titular Big Bang. Which in itself requires lots and lots of ontological paradox, as he shows up from the future to tell himself in the past what to do. Rory ends up guarding Amy in the Pandorica for 2000 years, while The Doctor pops in and meets him a minute later. The Doctor goes back and meets up with Amy in The Time of Angels, giving himself a loophole to reenter the universe through a traditional wedding rhyme. Then the universe restarts and everyone dances to Queen while Amy and Rory get married and River trades some riddles with The Doctor.
Even by Doctor Who standards, none of this makes too much sense if you think about it. So don't think about it. Just enjoy how weird and loopy it gets, and refresh your scorecard about all the events in time and space. The only mystery not solved in this episode, and still extant some years later, is how the TARDIS blew up in the first place. There is some thought that this won't be solved until the end of The Eleventh Doctor's run.
And if this two part finale pushed the boundaries of Doctor Who, it will still be short of the pyrotechnics of the series six two part opener, The Impossible Astronaut/The Day of the Moon.