There are many different aspects of this excellent short story that you could focus on to bring out the mystery, horror and the elements of the supernatural. Key to focus on is the way that W. W. Jacobs, in this story, gives us a horrific retelling of the traditional "three wishes" tale that can be found all over the world. The difference between this tale and its many variations is the way that it incorporates terror and suspense in its narration. One key element that makes this story terrifying is the description of the monkey's paw and the way that it is given a life of its own. Note what happens directly after Mr. White makes his wish for money:
A fine crash from the piano greeted the words, interrupted by a shuddering cry from the old man. His wife and son ran toward him.
"It moved," he cried, with a glance of disgust at the object as it lay on the floor. "As I wished it twisted in my hand like a snake."
Note the way in which the sound of the piano is used to foreshadow the disaster that is going to befall the White family. The crash is emotionally jarring, suggesting that something bad will happen. The crash is also surprising, indicating that the money will be provided but in a way that nobody can foresee. Note too the way that the monkey's paw twisted "like a snake." Snakes are creatures that we associate with evil, and so horror is injected through this comparison.
Through this example and many others, mystery, the supernatural and horror are introduced into this excellent tale.