An effective beginning for a scary story either sets the scene, developing an atmosphere of fear, or drops the reader straight into the terrifying action. You might begin by describing the haunted setting your characters have arrived at, or by introducing a discovery which heralds the start of the story's plot; you could even simply describe the local weather to establish a feeling of foreboding. A more action-packed beginning could feature characters being chased, or start with a shock opening.
Every story needs protagonists as well as the characters they'll meet; in a scary tale, this latter group often includes frightening and bizarre people, animals and even monsters. A horror story could feature an insane scientist or doctor conducting dangerous experiments, or perhaps the strange inhabitants of a forgotten town or castle. Animals such as black cats and dogs can be eerie in a scary story, while fictional creatures such as ghouls and goblins can all have a place. If you're stuck for inspiration for a monster, think of all the scary movies you've seen or books you've read, or even about the last Halloween party you went to.
A suitable location is vitally important to establishing the correct atmosphere for a scary story, and is also often integral to the plot; it might even be the story's setting which is haunted. Creepy old houses and graveyards are clichés of this genre, but you could still come up with a new twist on these locations. Think about other places too: buildings such as ruined churches and monasteries, castles and towers all make for suitably forbidding settings, while inns, harbors and ships are more unusual but still potentially effective. Even a deserted farmer's field can be frightening in a well-told tale.
Suspense is an important element in a scary story, and one you'll need to master to create an effective tale. As defined by the Short Stories Help Children website, suspense essentially means teasing the reader about what is going to happen, without giving too much away, so that she carries on reading your story. Gradually build up a plot, adding in incidents which create tension, and providing clues for readers, setting up as you do a suitably rewarding reveal at the end of the story. Foreshadowing later events by hinting at them earlier in the tale is another effective way of building suspense.