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. Ranks in the Church included Bishop, Cleric, Verger, and Colonel. The organisation itself was sometimes just called "the Clerics".

Members were organised into different faith/regiments, including the Anglican Marines. (DW: A Good Man Goes to War) The Papal Mainframe has been said to be the Headless Monks' and the Church's governor in separate sources.

Planets that were "Heaven neutral", like the Gamma Forest, fell outside the Church's jurisdiction. (DW: A Good Man Goes to War)

History
The Eleventh Doctor suggested that the Church was an evolution of the churches of Earth (DW: 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 went off in his blue box. Nonetheless he still told the Doctor he had faith in him when they were surrounded by Angels. (DW: The Time of Angels) A crack in space-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. (DW: Flesh and Stone)

Since the Doctor later negated the cracks, (DW: The Big Bang) the fates of the Clerics and the circumstances of the Angels' defeat is unclear.

The Church battled against Lava Snakes. ( DW : The Time of Angels )

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; at this time, the Church considered him a threat to humanity. 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. (DW: A Good Man Goes to War) The Clerics later helped the Silence capture River Song and place her in the astronaut suit. (DW: Closing Time)

Byzantium Expedition

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

DEMON'S RUN

 * Colonel Manton (leader)
 * Madame Kovarian (associate)
 * Headless monks (honourary allies)
 * Cleric Lorna Bucket
 * "Thin Man"
 * "Fat Man"
 * Lucas
 * Dominicus
 * Digger

Behind the scenes
According to The Doctor Who Encyclopaedia, a non-canonical REF book; at least one governing body was the Vatican. According to The Brilliant Book 2012, a non-canonical REF book; they also possessed their own Papal Mainframe.