The Headless Monks were a religious order who believed in listening to their hearts rather then their minds, to the point of having themselves beheaded.
As they were headless they did not register as living beings. The monks were also supposedly incapable of being fooled. Despite their lack of heads they seemed to be able to see well, or at least were able to detect where other beings were. They could also fire energy from their hands.
They could also be killed just as easily as most other humanoids, in this case with energy blasts or sword wounds.
The Headless Monks were very religious, although it's not explicitly stated what deity they worshipped. They believed in listening to their hearts rather than their minds, so they beheaded themselves with an unnamed, box-like device. If another group allied with the monks, they were expected to provide at least one of their number to join the monks.
The group followed an apparently female 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 apparently had many chants, including an attack chant. The Headless Monks were present at Demons Run when they fought the Eleventh Doctor and his allies. (DW: A Good Man Goes to War) The Delirium Archive was said to be their final resting place. (DW: The Time of Angels)