The answer to the first question was one of Matt Smith's biggest storylines in Doctor Who, and here's what the entire mystery was eventually revealed to mean. Season 5 of the revived Doctor Who saw wholesale changes with Matt Smith's Eleventh Doctor replacing David Tennant and incoming showrunner, Steven Moffat, taking over from Russell T. Davies. While Moffat's tenure elevated Doctor Who to greater success, some seasons were criticized for being too convoluted, with arcs that spanned multiple seasons and the "first question" is a prime example of this.
The mystery of the first question covers Matt Smith's entire Doctor Who stint, from his debut appearance in the home of Amy Pond to his final moments before regenerating into a Scotsman with bushy eyebrows and a permanent frown. The story behind this supposedly ancient question is told in a typically timey-wimey way, featuring paradoxes and red herrings, but is somewhat simpler to understand when viewed in hindsight.
When the modern incarnation of Doctor Who burst onto screens with Christopher Eccleston's Ninth Doctor in 2005, longtime fans were shocked to discover that Gallifrey and the Time Lords had been wiped out in the intervening years due to an unseen event called the Time War. It subsequently transpires that the Doctor was instrumental in ending the Time War, but did so by destroying Time Lord and Dalek alike.
After regenerating into his Eleventh form, the Doctor crashes into the life of Karen Gillan's Amy Pond and discovers a crack in time causing problems in her childhood bedroom. The Doctor resolves to investigate this crack, and the many others like it scattered throughout the universe, with the help of an adult Amy. The first mention of a "question" comes from new villains, the Silence, whose motto seems to be "silence will fall when the question is asked." Although the Doctor escapes the Silence's attempts to kill him, blue head in a box, Dorian Maldovar, reminds Eleven that the question has yet to be solved and claims it to be one of the oldest and most feared in history, also offering a better translation: "Doctor who?"
In the 50th anniversary special featuring David Tennant's Tenth Doctor, the audience, as well as the Time Lord himself, learn that Gallifrey was not actually destroyed, but confined to a separate pocket universe in order to put an end to the Time War, and this revelation leads directly to the first question's answer.
Matt Smith's final Doctor Who episode reveals the truth behind the Silence, the question and the cracks in time. On the planet Trenzalore, a crack had appeared and on the other side of this rupture were the Time Lords, wishing to know whether it was safe to return from their pocket universe. To ascertain whether the coast was clear, the residents of Gallifrey echoed a question throughout time and space that only the Doctor, the last of their kind, could answer. This question was "Doctor who?"
The Gallifreyans needed the Doctor to tell them whether it was safe to emerge and the only way to truly verify the Doctor's identity was to ask for his real name. Fearing that the return of the Time Lords would trigger a second Time War, the Silence were established to stop the question being answered and this is why they initially sought to kill the Doctor.
Eventually, the Doctor found the crack on Trenzalore, but most of the Doctor Who universe's villains followed, descending upon the planet. The Eleventh Doctor stayed on Trenzalore for hundreds of years, protecting the crack in time but refusing to answer the question and set the Time Lords free, as doing so would trigger a huge conflict with the waiting baddies. During this time, the Silence decided to aid the Doctor, but a rebel offshoot group decided to take more drastic measures and traveled back in time to destroy the Doctor's TARDIS, as seen in "The Big Bang." Paradoxically, this explosion is the very cause of the cracks in time.
Ultimately, the first question goes unanswered, with Clara Oswald insisting the Doctor's real name should actually be nothing more than just "the Doctor," thereby convincing the Time Lords to give their last remaining hero a new regeneration cycle, and leaving Gallifrey to find a new way back home.
Doctor Who season 12 is set to premiere in 2020 on the BBC.