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/must 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. (TV: Let's Kill Hitler) 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" apparently came to pass when the Eleventh Doctor and several companions were drawn to Trenzalore by the Great Intelligence. There, 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 unless he revealed the answer; however he did not need to as River Song, due to being noncorporeal 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 time and space, used as a code by the Time Lords on Gallifrey who were sealed inside a pocket universe. They were transmitting this message to every point in existence at once as a code word: when the Doctor would speak his true name, the Time Lords would know it was safe to return. The Silence, an offshoot of the Papal Mainframe, decided that if the Doctor were to return Gallifrey to the universe, the Time War would begin anew, and so they vowed to ensure that he never spoke his name, answering the oldest question in the universe. (TV: The Time of the Doctor)