Order of the Headless
The Headless Monks were a religious order who believed in listening to their hearts rather than their minds, to the point of having themselves beheaded.
Biology
As they were headless, the Headless Monks did not register as living beings. They were also supposedly incapable of being fooled or feeling fear. Despite their lack of heads, the Monks were able to detect where other beings were. They could also fire energy blasts from their hands, similar to the energy of the swords they carried.
They could be killed just as easily as most other humanoids, with energy blasts, guns, or sword wounds. They also had some form of vocal ability, as they could chant. (DW: A Good Man Goes to War)
Culture
The Headless Monks were very religious. They believed in listening to their hearts rather than their minds, so they beheaded themselves, and placed their heads inside an unnamed, box-like device. If another group allied with the monks, they were expected to provide at least one of their members to join the monks.
It was a Level One Heresy, punishable by death, to lower the hood of a Headless Monk. The group followed an entity known as the Papal Mainframe. Although the monks considered it a heresy to reveal their headless state to outsiders, they were apparently able to do so under the orders of the Mainframe.
The Monks would perform an attack chant before certain fights. (DW: A Good Man Goes to War)
History
The Headless Monks joined an alliance, along with the Anglican Regiment of The Church and Madame Kovarian, dedicated to the defeat of the Doctor, this alliance set up a plan. Some time after Amy Pond's honeymoon, after her child was concieved, she was captured and replaced by a Ganger perfectly synched to the real Amy so that neither the Doctor nor Rory Williams, Amy's husband, would notice.
As far as Amy knew, she was continuing her adventures with the Doctor and Rory. In reality, Amy's pregnant body was taken to Demons Run in the 52nd century. The Headless Monks went to great effort to contain her and her unborn child. They paid Dorium Maldovar for delivering them the brain of a Judoon trooper, which contained security software they needed. (WC: Prequel (A Good Man Goes to War))
Somehow, the Doctor knew that Amy was a Ganger, and after traveling back to see the early stages of the Flesh the Gangers were created from, he destroyed her Ganger form. Amy awoke to find herself full-term pregnant in an alien hospital, with Kovarian looking over her. She went into labour, with Kovarian cruelly instructing her to push. (DW: The Almost People) After Amy's baby was delivered, Kovarian took her away and had a Flesh duplicate created as to serve as a decoy to keep the Doctor off her trail. The Doctor recruited an army of old friends and people who owed him debts to assault Demons Run and rescue Amy and her daughter. (DW: A Good Man Goes to War)
In accordance with the Headless Monks' alliance protocol, one Cleric was to be initiated into their religion. Though it was a Level One Heresy, punishable by death, to lower the hood of a Headless Monk, they gave permission to Colonel Manton to unveil the monks, to reveal knots of skin where the bases of their necks should be. The third monk removeed his own hood; revealing himself to be the Doctor in disguise. The Monks raised their swords and the marines raised their guns. The Doctor put up his hood as the lights went out. The lights come back on and the Doctor was nowhere to be seen. He revealed that he was disguised as one of the Monks; as his voice echoed his location cannot be pinpointed. A marine shot a hooded figure who turned out to be a real Monk. The Monks began to kill the marines in retaliation until the Colonel told everyone to ceasefire. With the Monks and Clerics unnarmed, the Doctor's own forces: an army of Silurians and Judoon led by Sontaran Commander Strax, suddenly appeared with their guns pointed at the marines. Commander Strax held the Colonel at gunpoint, claiming the base.
The Doctor had a group of Spitfires attack the base, disabling communications. Amy and her supposed child were rescued by Rory Williams. Colonel Manton was forced to surrender, he, Kovarian and the rest of the Clerics (bar Lorna Bucket) abandoned the base. However, the Headless Monks struck, they massacred most of the Silurians before they even noticed. The Monks then surrounded the Doctor's remaining allies, Rory, Dorium, Strax, Madame Vastra, Jenny and Lorna. After killing Dorium, a battle started. This resulted in all the Monks being killed, while Strax and Lorna where both mortally wounded and died shortly afterwards. It was later revealed that the child they saved was a Ganger, and in reality, Melody Pond was taken by Kovarian. (DW: A Good Man Goes to War)
The Delirium Archive was said to be the final resting place of the Headless Monks. (DW: The Time of Angels)