In literature, a symbol is something that represents something else.
A symbol is a person, a place, an activity or an object that stands for something beyond itself.
The item that seems to be a symbol can represent itself. For example, a rose may be a gift from a man to a woman and it is simply that: the gift of a rose. However, on a deeper, symbolic level, the rose may also have some hidden significance. Roses are often symbolic of love. Red roses are symbolic of passion. A white rose is symbolic of love or innocence. A rose might be symbolic of beauty, while the rose's thorn could infer that beauty can sometimes cause pain.
Symbolism is a form of figurative language, which should not be taken literally.
Figurative use of language is the use of words or phrases that [imply] a non-literal meaning, which does make sense or that could [also] be true.
As a culture, we unconsciously use symbols to represent other things all the time. A peace sign signifies peace, as does a dove. A dollar sign represents money. The eagle is symbolic of freedom or of the United States. Green often implies envy; red is sometimes symbolic of anger. Black is symbolic of death or evil. For instance, years ago in old cowboy movies, the good guys wore white hats, while the bad guys wore black hats. Someone accused of being a coward in one of these films was said to have a yellow stripe running down his back. Literally, there was no stripe; it was a figurative way of saying someone was fearful: afraid to risk his safety or his life.
It is not unusual to have to search for symbols in a story. One indication that symbolism may be used in a story is the repeated reference to an object. If the color red is used repeatedly, it's quite possible that it is being used symbolically. Animals are symbolic. The fox is associated with sneakiness. Snakes are often associated with evil. Birds are also symbolic. In "The Raven" by Edgar Allan Poe, the bird symbolizes loneliness, loss and death.
The use of symbols allows an author to share information on more than one level. While there may be one story being told on the surface, the use of symbolism provides deeper layers to a piece of literature, conveying another hidden message.
In Poe's short story "The Tell-tale Heart," the old man's eye is believed by many to be symbolic. On the surface it is the characteristic of the old man who is murdered in the story. However, repeated reference to the eye makes it symbolic. It could symbolize the narrator's inability to see the world clearly—hindered by his madness. It may symbolize death: the old man may have an inclination of the narrator's intent, even though his eye is clouded with age and/or infirmity. The eye might also be symbolic of truth, or vigilance (watchfulness). In this case, the old man's eye may symbolize the narrator's inability to escape undetected.
Symbolism is subjective in nature. If you are searching for symbolism in a piece of literature, look for objects, colors, etc., that are referred to continually. Search for the symbolic meaning and then draw a logical correlation between the action of the story and the significance of the symbol in providing deeper insights into the author's tale. There is some poetic license that may be taken in searching for symbolism. However, you must collect information that supports your reasoning. If the support is not there, you cannot make a case for the use of symbols: for symbols serve to tell a story on a level different—perhaps hidden—from the story on the surface.
The Language of Literature. McDougal, Littel and Company: Illinois, 2006.