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 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.
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.
The Monks would perform an attack chant before certain fights.
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.