The Question was, according to the Teselecta's computer, "the first question, the oldest question in the universe, hidden in plain sight". It was the organising principle for the religious movement known as the Silence, who believed that "silence will fall when the Question is asked". Accordingly, they called themselves the "Academy of the Question".
More specifically, the Silence believed that
... on the Fields of Trenzalore, at the fall of the Eleventh, when no living creature can speak falsely or fail to answer, a question will be asked — a question that must never ever be answered: "Doctor who?"
The Silence planned to kill the Doctor, whom they believed was the only one who knew the answer to the Question. At one point, they attempted to do this by turning Melody Pond into a weapon to kill the Doctor. Her first attempt was in 1938 Berlin and then again in 2011 on the beach of Lake Silencio. Although this attempt also failed, the Silence (as well as the rest of the universe) believed it was successful. Unbeknownst to the Silence, the Doctor had used the Teselecta to fake his death at Lake Silencio. (TV: The Impossible Astronaut, Let's Kill Hitler, The Wedding of River Song)
Dorium Maldovar told the Doctor that "silence will fall when the question is asked" was a poor translation. A better translation would be "silence must fall..." Although the Teselecta's databases were unable to identify the Question, Dorium claimed to know it. According to him, the Question was actually, "Doctor who?" (TV: Let's Kill Hitler, The Wedding of River Song)
Some of the events described in Maldovar's "prophecy" seemingly came to pass when the Eleventh Doctor and several companions were drawn to Trenzalore by the Great Intelligence, but these events were not what the "prophecy" actually referred to. On Trenzalore, a future version of the Doctor's TARDIS was found, but the doors were sealed and the only way to enter was by answering the question, "Doctor who?" — uttering the Doctor's true name. The Great Intelligence threatened the lives of the Doctor's companions in order to force him to reveal the answer; however, he did not need to as River Song, due to being a non-corporeal data ghost at this point in time, and who was aware of the Doctor's true name, (TV: Silence in the Library) was able to utter it from outside the TARDIS, opening the doors and preventing the Doctor from having to reveal the secret. (TV: The Name of the Doctor)
It was revealed that the question was actually a transmission that traversed all of time and space, making it the "oldest question in the universe," and was used as a code by the Time Lords on Gallifrey who were sealed inside a pocket universe. They were transmitting the message through a crack in time on the planet Trenzalore, asking the Doctor to speak his true name in order to let them know that it was safe to emerge from the pocket universe. Along with the Question, the Time Lords also transmitted a Truth Field which enveloped the town of Christmas so that the Doctor could not lie if he were to answer the Question. This prompted a siege on the planet by a host of alien races who wanted to prevent the start of another time war, but they were held off for centuries by a combination of the Doctor and a force field that had been put in place by the Papal Mainframe. In response to this, Tasha Lem commissioned a faith change and established the Church of the Silence, dedicating it to the cause of maintaining the Doctor's silence and keeping him from speaking his name. However, the Kovarian chapter, an offshoot of the Church, went to more extreme measures in their attempts to prevent the Doctor from speaking his name and releasing the Time Lords, travelling back along his timeline and attempting to kill him retroactively. One of their attempts on the Doctor's life involved the destruction of his TARDIS, although this attempt backfired on them and ended up creating the very cracks in time through which the Time Lords were calling through. (TV: The Time of the Doctor, The Pandorica Opens)
The conflict on Trenzalore eventually boiled down to the point that only the Daleks remained at war with the Church and the Doctor. The Doctor, being out of regenerations and convinced of his foreseen death on the planet, intended to let the Daleks kill him, but Clara Oswald believed that the future could be changed and implored the Time Lords to help the Doctor. Ultimately, she became the one to answer the Question, although she stated that the Doctor's real name was simply "the Doctor," and that the meaning behind that answer was a sufficient-enough reason for the Time Lords to give him their support, which they then did by changing the future and granting the Doctor a brand new regenerative cycle, thereby allowing him to defeat the Daleks and survive the battle. (TV: The Time of the Doctor)