The most significant metaphor in the story is the hunter versus the hunted.
A metaphor is when an idea stands for something else. The biggest extended metaphor in the story is voiced by Rainsford at the beginning of the story. It is part of the conversation that he has with Whitney about whether or not animals feel. Rainsford’s position is that they do not.
"Nonsense," laughed Rainsford. "This hot weather is making you soft, Whitney. Be a realist. The world is made up of two classes--the hunters and the huntees. Luckily, you and I are hunters.
This is part of the overall metaphor in the story of the weak versus the strong, or the hunter versus the hunted. Rainsford and Zaroff act out this metaphor in their interaction when Zaroff forces Rainsford to play the game.
Generally speaking, this story is full of figurative language. It helps create a picture in the reader’s mind. While an extended metaphor is a big concept used throughout the story, you will find several smaller metaphors used throughout. Do not confuse them with similes (“It's like moist black velvet."). A simile is an indirect comparison. A metaphor does not use "like" or "as."
Compare that to a metaphor.
“It's so dark," he thought, "that I could sleep without closing my eyes; the night would be my eyelids--"
When you say “night would be my eyelids” it is not literal. Night is not literally an eyelid. It just means it is very dark. It is a way of setting the mood, and showing Rainford’s emotional state.
Consider this metaphor too.
The lights of the yacht became faint and ever-vanishing fireflies; then they were blotted out entirely by the night.
In this case, the lights of the ship are compared to fireflies. It helps you picture what they look like, but again, also helps to capture Rainsford’s state of mind as he watched the ship leave him in the water. He knew that he was left behind, and he feared he might die if he did not make it to shore.
In the case of both extended metaphors and the smaller metaphors develop Rainsford's state of mind throughout the story. As he goes from being the hunter on the ship to the hunted in Zaroff's clutches, he is at first afraid, and then soon finds his nerve.