The Church

The Church was a faith-based military organisation primarily operating in the 51st century. It was responsible for interstellar security, protecting human populations, and special covert operations. Instead of regular military ranks, the Church operated under more religious offices such as Bishop, Cleric and Verger.

They are broken down into multiple different faith/regiments, including the Anglican Regiment and the Headless Monks, with a Papal Mainframe (refered to as "her"/"she") in some position of authority. (DW: A Good Man Goes to War)

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

Father Octavian recruited River Song for a covert mission in which she infiltrated the Galaxy class Starliner, Byzantium as it contained a Weeping Angel. She was able to gain the Doctor's attention which lead to him rescuing her and accompanying her to Alfava Metraxis, where the Byzantium had crashed. There they united with Octavian. Octavian led his men into the Maze of the Dead below the Byzantium, Christian, Angelo and Bob were soon killed by the Angel. The Doctor assured him they could not have been saved, but Octavian reminded 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. Although the Doctor, Amy and River were able to survive and send the Weeping Angels falling into the crack. 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 negated the cracks, (DW: The Big Bang) the fates of the Clerics and the circumstances of the Angels defeat is unclear.

Either before or after this, a multi-faith force of the Church was part of an operation to capture the infant Melody Pond in the hopes of using her as a weapon against the Doctor - they considered a dangerous threat to humanity. The Doctor was able to neutralise the entire force at their Demon's Run base without killing anyone (though he got the Anglicans and the Headless Monks to fight for a brief period). However, this was part of a trap - the Anglicans may not have been aware - and the infant remained in their hands. (DW: A Good Man Goes to War)