The Church

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 Bishop, Cleric, Verger, and Colonel. Members of this organisation were sometimes called "the Clerics."

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 a 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)

The Church battled lava snakes. (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)

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 to humanity, presumably through the Silence's post-hypnotic suggestions. The Eleventh Doctor neutralised the entire force at their Demon's 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

 * Bishop Second Class Octavian
 * Cleric Christian
 * Cleric Angelo
 * Cleric Bob
 * Cleric Marco
 * Cleric Pedro
 * Cleric Phillip
 * Cleric Crispin

Demon's Run

 * Colonel Manton (leader)
 * Madame Kovarian (associate)
 * Order of the Headless (honourary allies)
 * Lorna Bucket
 * "The Thin One"
 * "The Fat One"
 * Lucas
 * Dominicus
 * Digger

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, also a book that contains non-narrative based information, they also possessed their own Papal Mainframe.