According to Dictionary.com, an allegory is "a representation of an abstract or spiritual meaning through concrete or material forms" or else "figurative treatment of one subject under the guise of another." Thus, an allegory is a verbal or written statement in which a tangible reality is used to express something less tangible. Allegories often take the form of a story used to express a lesson, warning or even a request.
Dictionary.com defines symbolism as "the practice of representing things by symbols, or of investing things with a symbolic meaning or character." Symbolism occurs when an object or character is invested with a particular meaning that it does not, by definition, have. For instance, the dove might symbolize peace; however, nothing in and of the dove itself represents peace --- the meaning is attributed to the dove symbolically.
Allegories and symbolism are related. Both are meant to convey a meaning that is different than the thing directly presented. However, while allegories are stories meant to convey a lesson or similar messages, symbols form a broader field of meaning and are objects --- not stories --- that embody some particular meaning.
As noted, the dove is a common symbol representing peace. Other symbols include religious symbols, such as the crucifix or crescent, which represent Christianity and Islam. The symbol of a red cross is internationally recognized as representing the Red Cross organization and often stands to represent a medical or humanitarian service that is meant to be free from military or political attack or interference. In math, the minus sign is a symbol that usually means "subtract."
One of the most famous allegories is Plato's "Allegory of the Cave." Plato uses this story to explain how some segments of the population are ignorant while others have come to "see the light." In art, allegories are often used, such as the use of a skull, hourglass or dead fruit, to tell the story of death and represent a reminder that human beings are mortal creatures.