Remember that a symbol in literature is defined as an object, action or person that stands for something beyond their literal meaning. So, thinking about symbols, a dove is a common symbol for peace or a cross for Christianity. Both have their literal meaning, and then also a bigger and wider meaning.
Thinking about this excellent horror story then, it is clear that one thing that acts symbolically is the monkey's paw itself. When the Sergeant Major tells us of the background of the paw and how it was created, this symbolic significance becomes clear:
"It had a spell put on it by an old fakir," said the sergeant major, "a very holy man."
"He wanted to show that fate ruled people's lives, and that those who interfered with it did so to their sorrow. He put a spell on it so that three separate men could each have three wishes from it."
This monkey's paw, therefore, has a far more symbolic significance than simply being a paw of a monkey. The fakir created it as a symbol of man's desire to play with his fate and his belief that he can meddle with the way things are going to work out. It is this symbolism that is understood and grasped by Mr. White by the end of the tale but not by Mrs. White, with almost tragic consequences.