In "The Monkey's Paw" Jacobs creates a sense of horror in several ways. First, he opens the story with a scene of calm domesticity. This gives us something to threaten and upset.
Second, he gives a symbolic foreshadowing in the first paragraph, when he speaks of the characters playing chess and putting their pieces into "sharp and unnecessary perils." What's more, the old woman watching can see that these moves will lead to danger, but no one listens.
Third, when the story develops, the soldier who has seen a lot doesn't want to share the story of the monkey's paw; he warns them, just as the old woman warned the chess players. This continues when he tells them it is better to let the paw burn, and that the other man's third wish was for death.
Fourth, as the wishes play out, the atmosphere is developed by the wishes going wrong, and by the couple being unable to resist the power of the magic at the same time they are realizing things will always go wrong from it.