The Headless Monks, also known as the Order of the Headless, were a religious order who believed in listening to their hearts rather than their minds, to the point of having themselves beheaded.
As they were headless, the Headless Monks did not register as living beings. They were also supposedly incapable of being fooled or surprised or of feeling fear. Despite their lack of heads, the Monks were able to detect where other beings were. They could fire energy blasts from their hands, as well as channelling that energy into 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, though this may have been a psychic ability. (TV: A Good Man Goes to War)
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 attempt 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.
Not all inductees to the Order of the Headless had their heads kept in boxes. The boxes were apparently reserved for the very wealthy, while the rest of the monks' heads were left to rot on shelves in a cave full of skulls called the Seventh Transept. Despite this, the skulls were still 'alive' in a sense, capable of moving about and attacking others. The heads kept in ceremonial boxes were even better preserved: Dorium Maldovar was still very active and in full possession of his memories and personality when visited by the Eleventh Doctor. (TV: The Wedding of River Song)
The Order of the Headless joined an alliance presumably led by the Silence, along with the Anglican Regiment of the Church and Madame Kovarian, dedicated to the defeat of the Doctor. Some time after Amy Pond's honeymoon, after her child was conceived, she was captured and replaced by a Ganger perfectly synced to the real Amy so that neither the Eleventh 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 Demon's Run in the 52nd century. The Headless Monks went to great effort to contain her and her unborn child. (TV: A Good Man Goes to War) 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))
In accordance with the Headless Monks' alliance protocol, one Cleric was to be initiated into their religion. Later, during a speech by Colonel Manton, he was allowed to unveil the monks to reveal knots of skin where the bases of their necks should be; the Cleric had been unaware that this would happen to him.
The Doctor revealed himself to be disguised as a Monk, in order to panic the Clerics into attacking them. The Monks began to kill the marines in retaliation until Manton told everyone to cease fire, at which point they were surrounded by the Doctor's own forces, an army of Silurians and Judoon led by Sontaran Commander Strax. The Doctor believed he'd captured and driven off the entire force — however, he was unaware the Headless Monks did not register as alive and that they'd kept most of their number hidden, as part of a trap. They massacred most of the Silurians before they were even noticed and took out the Doctor's allies Strax, Lorna Bucket, and Dorium (the last converted into a Headless) before they were killed. (TV: A Good Man Goes to War)
The Eleventh Doctor later discovered that the Headless Monks stored the heads of their followers in the catacombs under their temple, in the Seventh Transept. Despite having no bodies and the flesh around their skulls having rotted away, the skulls were still alive, although bestial in nature; they would attempt to eat anything that wandered into the catacombs, including rats and people. Gantok took the Doctor here to meet the severed head of Dorium Maldovar. Dorium's head turned out to be fully alive and still sentient, a privilege affordable only by the wealthy. (TV: The Wedding of River Song)