The Silence

The Silence was a religious order who considered themselves the "Sentinels of History". They tried to kill the Doctor to prevent the fruition of a specific prophecy, which stated: "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."

This question was "The First Question, the oldest question in the Universe, that must never be answered, hidden in plain sight." Dorium Maldovar told the Eleventh Doctor that the question was: "Doctor Who?", which was a question the Doctor had been apparently running from his entire life.

Organisation
The Silence was led by a race of aliens who would be erased from your memory after you looked away from them (known as Silents) who used post-hypnotic suggestion to manipulate other species into doing their bidding. The Silents also employed human or humanoid agents to carry out specific tasks. These agents were often fitted with Eye Drives which allowed them to remember the Silents. (TV: A Good Man Goes to War, The Wedding of River Song)

History
The Silence allied with Madame Kovarian, the Church and the Order of the Headless. They were behind the Battle of Demon's Run and the kidnapping and brainwashing of the infant River Song, then known as Melody Pond. (TV: A Good Man Goes to War) She later made several attempts on the Eleventh Doctor's life. Ultimately, she kissed him with poisoned lipstick. She later broke free of her conditioning and revived him using the remainder of her regenerative energy. (TV: Let's Kill Hitler)

The alien leaders of the Silence were encountered by the Doctor in 1969 America. The Silence had been on Earth since humanity's beginning, secretly manipulating humanity and its development for their own ends.

It was implied they were responsible for the Moon landing because they needed a special spacesuit for Melody Pond. The Silence's occupation of Earth was broken when the Doctor issued a post-hypnotic command via the Apollo 11 footage to the entire human race, instructing them to kill the Silents on sight. (TV: Day of the Moon)

The Silence tracked down River Song to the Luna University, where they kidnapped her and forced her into an astronaut suit as part of their plan to kill the Doctor. (TV: Closing Time). One Silent was present near Lake Silencio in Utah to observe the Doctor, River, Amy and Rory during the events of the Doctor's death. (TV: The Impossible Astronaut) Their plan was foiled when the Doctor convinced the crew of the Teselecta to disguise their ship as him and take his place, tricking the Silence into believing they had killed the Doctor at a fixed point in time. The fixed point was rather, the Teselecta's destruction and the deception itself. (TV: The Wedding of River Song)

Ultimately, the prophecy that the Silence were attempting to prevent came to pass, when The Great Intelligence asked the Question at Trenzalore. Answering the question allowed the Intelligence to gain access to the Doctor's tomb and rewrite his entire timeline, although the damage caused by the Intelligence was later undone by Clara Oswald, who sacrificed herself by following the Great Intelligence into the timeline. (TV: The Name of the Doctor)