Pearl in The Scarlet Letter is not so much the embodiment of sin as she is the product of sin. Through the Reverend's and Hester's digression, something beautiful is born. It is like a pearl that is formed in a clam from a single grain of dirt or sand. The character, Pearl, is something wondrous that is made from something "ugly" like sin. Pearl represents an innocence not tainted by sin, and she is a redeeming character for Hester.
Many times historical people or literary characters become symbols of a trait or value of society. For example, Abraham Lincoln represents honesty; Huck Finn represents individuality. You can probably name a quality many people in your own life represent--a mother symbolizes love, a friend symbolizes acceptance, or a teacher symbolizes wisdom. By making a character or person a symbol of an abstract quality we value in society, an author establishes a relationship between the reader and character so we can better empathize with and understand the character.