The term "horror" describes a kind of grab-bag of horrible, black-aligned beings-that-should-not-be. Horrors can be birthed from any number of grim sources, including the basements of parents' homes, the back rooms of tabletop game shops, or the student union on Wednesday nights.
Whenever one of these Creatures (originally from the Odyssey Block) comes into play, another card (of a type specified in the Horror's rules text) in play is removed. When the creature leaves play, the card it "replaced" is returned to play. Combos have arisen from these abilities, most notably taking advantage of a loophole regarding Faceless Butcher's.
This particular brand of Horror is not unified by a specific ability, but rather a common flavor. They are all related to the stories of the plane of Phyrexia or the Phyrexian invasion. While there is not one prevalent ability, most of their abilities do some sort of harm to the controller of the Horror.