"We can only accept huntsmen whose heads have parted company with their bodies. You will appreciate that it would be impossible otherwise for members to participate in hunt activities such as Horseback Head-Juggling and Head Polo."
-Sir Patrick Delaney-Podmore in a letter to Nearly Headless Nick
The Headless Hunt is a group of a dozen decapitated ghosts who ride ghost horses and engage in such activities as Horseback Head-Juggling, Head Polo, and Head Hockey. The Hunt is led by Sir Patrick Delaney-Podmore, who maintains a strict policy to only accept huntsmen whose heads have completely separated from their bodies (CS8).
In 1992 the Headless Hunt regretfully rejects Nearly Headless Nick’s application to join via a rejection letter from Sir Patrick, showing that ghosts can communicate by letter. The Headless Hunt crashes Nick’s deathday party by galloping onto the dance floor and playing Head Hockey while Nick attempts to make a speech (CS8).
The Headless Hunt is last seen at the Battle of Hogwarts loping along the seventh floor corridor yelling battle cries (DH31).
Sir Patrick's letter to Nearly Headless Nick shows ghosts can communicate via letter.
Although Nearly Headless Nick tells Harry only wizards can come back as ghosts (OP38), the headless huntsmen ride ghost horses, indicating that creatures other than wizards can return from the dead in ghost form. It is unclear, though, whether the horses were bound to the wizards in life, therefore somehow binding them in the afterlife as well; if the horses are magical themselves; or if other rules dictating ghostliness govern non-humans.
That headless huntsmen's heads and bodies can function when detached provides another example of ghosts being less restricted in movement and bodily functions in the afterlife than when living.
The Headless Hunt's contribution to the Battle of Hogwarts is unknown. It is possible they could have assisted by distracting enemies with noise, acting as a translucent barrier to block visibility, and dousing enemies with a cold sensation.
The Headless Hunt is likely an homage to the headless horseman motif popular in European folklore for centuries. In Scottish lore, a horseman named Ewen was decapitated in a battle at Glen Cainnir on the Isle of Mull and, along with his horse, continues to haunt the area.