Nathaniel Hawthorne's "Young Goodman Brown" is full of symbols; in fact, it is a great example of an allegory, a literary work that operates on both literal and symbolic levels. You could choose any number of symbols from the story—the man Brown meets in the woods, the names of the characters, the satanic ceremony in the woods—but I want to focus on Faith's pink ribbon as a good example of how a symbol conveys the concerns of the work as a whole.
At the start of the story, Goodman Brown leaves his young wife Faith, aptly-named, as Brown thinks of her as representative of total innocence and purity. The note of her "pink ribbons" at the beginning of the story doesn't stand out much, but later, when the protagonist see the ribbon again in the forest, it takes on much greater significance.
As the story progresses, Young Goodman Brown meets a man in the forest who represents the devil. This man reveals all sorts of dirty secrets of Brown's strict Puritan town. Even Brown's religion teacher is friendly with the devil. This symbolizes that all people have some good and some evil, or rather than no one is perfect. A more cynical reading could even say that it means Puritans are hypocritical because they pretend to be good and pure but are actually sinners who are just good at hiding their flaws. Eventually, Brown reaches a satanic ceremony in the woods that seems to mirror a baptism where new members are being initiated into the church. Here, though, the new members are pledging allegiance to Satan. One of the new members is, of course, Faith, Brown's innocent wife. This is a huge shock to Brown, but the pivotal moment occurs when Faith's ribbon reappears:
The cry of grief, rage, and terror, was yet piercing the night, when the unhappy husband held his breath for a response. There was a scream, drowned immediately in a louder murmur of voices, fading into far-off laughter, as the dark cloud swept away, leaving the clear and silent sky above Goodman Brown. But something fluttered lightly down through the air, and caught on the branch of a tree. The young man seized it, and beheld a pink ribbon.
“My Faith is gone!” cried he, after one stupefied moment. “There is no good on earth; and sin is but a name. Come, devil! for to thee is this world given.”
When Brown realizes that Faith's pink ribbon has fluttered down to him, he takes it as a sign that his wife has gone to "the dark side" and is no longer innocent. The pink ribbon here could be a symbol of that innocence that is now lost. Along with her innocence, Goodman Brown's "Faith" in the good of his fellow people and in religion or God at large has been destroyed. Brown claims the devil rules the world now.
I think, however, that the color of Faith's ribbon is significant here. Pink...
is a mixture of red and white. In literature, white is often associated with purity and innocence. Red in our story here, as in other Hawthorne works (The Scarlet Letter) represents guilt or sin, and is associated with Satan. Faith's ribbon is a symbol of her larger persona—she is a mixture of innocence (good) and sin (evil), just like everyone else. Her ritual in the woods is almost like an introduction to adult life rather than a promise to worship Satan. This is why Faith and everyone else in the town is acting totally normal the next morning, while Brown is completely devastated by this knowledge. Faith's ribbons were pink all along; Goodman Brown simply did not yet know the truth that all people are complex and no one is purely good.