Defining a genre, any genre, is difficult. Particularly when that genre is as diverse as horror. My definition is no absolute, it is what works for me and may work for you.
horror encompasses a wide slate of stories and films, everything from the purely psychological such as Silence of the Lambs, to the most fantastic premises such as kaiju movies like Godzilla. (Or Gojira is you are a purist.) trying to find one definition, one element that binds all of these together is perhaps a fools errand. In fact my working definition of horror actually excludes the purely psychological sub-genre, as I tend to place thrillers in their own genre distinct from horror. Not better, not worse, just their own genre.
So what is my definition? Horror is the field of story telling where the essential nature of the story is that the world is not as the protagonist, and by extension the viewer/reader expects it to be. The rules that the protagonists have accepted are reality are undone and can never be assumed to be true again.
Consider ghosts. We live in a materialistic world where the belief in ghosts is consider eccentric. They are not considered a part of reality and certainly not reality as described by either the physical or biological sciences. If someone comes home to a haunting that is frightening because it violates accepted reality.
Horror becomes adventure when the new reality is the accepted paradigm. Hence Alien is a horror film, Aliens is an adventure film. It also explains the difficulty in craft an genuinely good horror film fro well mined material, vampires, werewolves, zombies and the like have well known rules and that very existence of well known rules undercuts and destroys the sensation of horror.
Now for some horror is about violence, bloodletting, and if we are talking film, explicit effects, but for me that is mere shock and falls short of true horror. Horror to me is a very cerebral sensation. the best stories and films leaves me with a sense of unease that haunts my thoughts well after the tale is told. It the unsettled image of a man sleeping every night in the bed with the ghost of his wife. It is the image of a mother, son, and a VCR on their way to find someone to pass the curse to, to kill. It is the old house on the hill and its power over a poor young woman weak of spirit. These are not images of violence, but they have a violence of the soul in them. To me that is the most powerful horror