What does Faith symbolize in "Young Goodman Brown"?

Nathaniel Hawthorne's "Young Goodman Brown" is a complex tale of a young Puritan man's decision to leave his wife and village and walk into the surrounding forest. Although the reason for this dangerous journey is ambiguously described first as "his present evil purpose," we learn later that he is taking a walk on the dark side to test his faith in his Puritan belief system, accompanied by a guide who turns out to be Satan. The journey, of course, may actually be taking place in his troubled mind in the form of a dream vision, but for Goodman Brown, the experience is real.
Young Goodman Brown's wife is Faith, whose name serves both to denote her role as wife and as an emblem of a 17th century Puritan's belief system. When she attempts to dissuade Goodman Brown from his journey, we learn that she, too, is troubled:

"Dearest heart," whispered she, softly and rather sadly, when her lips were close to his ear, "pr'ythee, put off your journey until sunrise, and sleep in your own bed to-night. A lone woman is troubled with such dreams and such thoughts, that she's afeard of herself, sometimes. Pray, tarry with me this night, dear husband, of all nights in the year!"

Faith refers here to the Puritan belief that Satan is able to visit dreamers and tempt them into unrighteous behavior because their will power is weakened during sleep, and Faith is expressing her concern that, if left alone, she might succumb to temptation. Hawthorne is cleverly sowing seeds of doubt about Faith's faith before Brown even begins to test his own faith. Hawthorne also implies, with "all the nights in the year," that the journey is taking place on All Hallow's Eve, a night when Satan might be more active than usual. Goodman Brown, however, believes that Faith's belief system is stronger than his, and he assures himself that "Well; she's a blessed angel on earth; and after this one night, I'll cling to her skirts and follow her to Heaven."
The story culminates at a witches' ceremony in which Goodman Brown believes he encounters Faith, and in an attempt to save her, he cries,

"Faith! Faith!" cried the husband. "Look up to Heaven, and resist the Wicked One!"

In Brown's imagination, his faith and his wife Faith are in equal jeopardy during Brown's experience, and Hawthorne creates additional ambiguity when he notes that "whether Faith obeyed he knew not."
Goodman Brown's wife Faith is, despite the dream vision Brown experiences, the embodiment of a conventional Puritan belief system. Brown's challenging of that belief system, however, is so pervasive that he imputes to Faith his own loss of faith, and, as Hawthorne describes Brown's life following this experience, "they carved no hopeful verse upon his tombstone; for his dying hour was gloom."

From what I can see, you are referring to Faith, Goodman Brown's wife. In the story, Faith symbolizes piety, decorum, and innocence. In short, she represents everything that is noble and virtuous in Goodman Brown's life.
In everything he does, Goodman Brown thinks of his wife; she is his lodestar or guide. When he leaves for his appointment, he feels a little guilty. After all, he is keeping his fearful errand a secret from his beloved, trusting wife. He berates himself and calls himself a "wretch." To him, Faith is a "blessed angel on earth."
Later, when his companion accuses him of being fifteen minutes late, Goodman Brown asserts that "Faith kept me back a while." Here, Faith represents Goodman Brown's conscience. Deep inside, Goodman Brown suspects that his foolhardy errand will not end well. Yet, he moves forward with a perverse excitement that is foreign to him.
Later he comes to regret his foray into the woods. When he sees the esteemed members of his congregation engaged in witchery, Goodman Brown is inconsolable. Had he stayed home, he would have been "purely and sweetly" sleeping "in the arms of Faith." Goodman Brown is in for a shock, however, when he later spies his beloved Faith in the congregation of "fiend worshippers." 
Because of his experience, Goodman Brown loses his trust in his religious leaders and begins to doubt even his beloved wife. The story reinforces the fragility of faith and the necessity of having a practical mindset about human nature.