The Church

The Church of the Papal Mainframe, later known as the Church of the Silence, simply known as the Church, was a religious and military organisation in the 51st and 52nd centuries. It was responsible for interstellar security, protection of human populations and special covert operations, and acted as the military force of the Papal Mainframe. Ranks in the Church included Mother Superious, Priest, Bishop, Cleric, and Colonel. Members of this organisation were sometimes called "the Clerics." The Church also included Confessional Priests, genetically engineered by the Church to allow people to confess their sins without remembering doing so, accomplished by fashioning the Priests so anyone looking at them would forget their encounter when they looked away.

There were Anglicans in the Church, some of whom were marines. (TV: A Good Man Goes to War)

Some planets were "heaven-neutral," like the Gamma Forests. It was considered unusual to find their inhabitants in the church military force. (TV: A Good Man Goes to War)

History
The Eleventh Doctor suggested that the Church was an evolution of the churches of Earth. (TV: The Time of Angels)

In the 51st century, the Church recruited River Song for a covert mission in which she infiltrated the Galaxy class Starliner Byzantium, which contained a Weeping Angel. She called in the Eleventh Doctor, who rescued her and accompanied her to Alfava Metraxis, where the Byzantium had crashed. There they met with Octavian's unit. Octavian led his men into the Maze of the Dead below the Byzantium. Clerics Christian, Angelo and Bob were soon killed by the Angels. The Doctor assured him they could not have been saved, but Octavian told him that he would be the one to tell their families while the Doctor flew off in his blue box. Nonetheless, he still told the Doctor he had faith in him when they were surrounded by Angels. (TV: The Time of Angels) A crack in space and time was discovered in the forest of the Byzantium. The Doctor, Amy and River survived and sent the Weeping Angels falling into the crack, erasing them from space and time. Octavian was killed by a Weeping Angel and the remainder of the Clerics were erased by the crack. (TV: Flesh and Stone)

When a mysterious message that filled "half the universe" with fear spread to every corner of space and time, the Papal Mainframe was the first ship to arrive at Trenzalore, the source of the message, and shielded it so that nobody could land. However, unknown to the church, the Weeping Angels had already arrived on planet.

Mother Superious Tasha Lem at some point summoned the Eleventh Doctor and Clara Oswald to the Mainframe in response to the message. Claiming ignorance as to its contents, she sent them down to the planet to discover the truth themselves.

After she discovered that the message was the First Question, sent by the Time Lords from their pocket universe, Tasha communicated to the Doctor that if he took action to save the Time Lords, he would cause the fleets above to destroy them. The Doctor refused to allow the Mainframe to destroy the planet and spent the next 300 years defending it. The Papal Mainframe also busied themselves with keeping the peace amongst the various species that remained in orbit.

Tasha officially changed the core tenets of the Papal Mainframe's constituent chapters so that they became dedicated to ensuring that silence would fall in response to the question, turning the Papal Mainframe into the Church of the Silence.

By the time of the Doctor and Clara's next visit to the Mainframe, it had been largely overrun by the Daleks, who converted most of the crew, including Tasha herself, into puppets. However, with the Doctor's assistance, Tasha managed to resist the Daleks' control and take back the Church, and spend the remainder of conflict coordinating the Papal Mainframe and the Silence against the Daleks. (TV: The Time of the Doctor)

In the 52nd century, a Church regiment and the Headless Monks were working together as part of the Silence's operation to capture the infant Melody Pond to use as a weapon against the Doctor; by this time the Church considered him a threat (TV: A Good Man Goes to War) because of the Doctor's ability to release the Time Lords from their pocket universe and start a new Time War along his personal future, which the Silence tried to prevent. (TV: The Time of the Doctor) The Eleventh Doctor neutralised the entire force at their Demons Run base without personally killing anyone (though he tricked the Clerics and the Headless Monks into shooting each other). However, the defeat of the Church was part of the Silence's trap. (TV: A Good Man Goes to War) The Clerics later helped the Silence capture River Song and place her in the astronaut suit. (TV: Closing Time)

Byzantium expedition
(TV: The Time of Angels, Flesh and Stone)
 * Bishop Second Class Octavian
 * Cleric Christian
 * Cleric Angelo
 * Cleric Bob
 * Cleric Marco
 * Cleric Pedro
 * Cleric Phillip
 * Cleric Crispin

Demons Run
(TV: A Good Man Goes to War)
 * Colonel Manton (leader)
 * Madame Kovarian (associate)
 * Order of the Headless (honorary allies)
 * Lorna Bucket
 * "The Thin One"
 * "The Fat One"
 * Lucas
 * Dominicus
 * Digger

Papal Maınframe
(TV: The Time of the Doctor)
 * Tasha Lem (leader)
 * Colonel Meme
 * Colonel Albero
 * Kovarian (rebelled)
 * Priests of the Papal Mainframe
 * Priests of the Kovarian Chapter (rebelled)

Behind the scenes
According to The Doctor Who Encyclopaedia, a book that contains non-narrative based information, at least one governing body was the Vatican. According to The Brilliant Book 2012, another book that contains non-narrative based information, they also possessed their own Papal Mainframe.