Headless horsemen are at-will-decapitated or willingly beheaded monsters that have the ability to remove and put back on their heads at complete will. They ride horses, of course, being "horsemen", and they take off their heads whenever they please.
Not much else is known about them.
There is a creature called a Dullahan (an irish demon without a head), which is said to be inspiration for the headless horsemen myth. A well known example of Headless Horsemen in fiction is Sleepy Hollow, where a Headless Horseman is featured as an antagonist.
Headless horsemen in Monster High
The only known one is Headless Headmistress Bloodgood, and she is the only headless horseman in the Monster High. She is fully willed at removing her head, favorite food is fresh scones with marmalade and a cup of tea, only "BFF" is when she is content with her own company and a good book, and is uncomfortable discussing her "freaky flaw". Though she does say her Freaky Flaw is when her body and head are in different locations. Though when her body and head are in separate areas, she always gets her head back. Her age, as she says: "Wise enough to be your grandmother - young enough to be your sister", although when Cleo/Toralei, Venus/Clawdeen, Draculaura/Robecca, and Jinafire Long attached to Lagoona are in the play rehearsal for when the school was founded in "Freaky Fusion", it is quite noted that Bloodgood was around back then, making her older than she says. She also has a pet nightmare horse named Nightmare, who is blue.