What were each of the characters sin and their salvation?
What other meanings does the Scarlet Letter "A" have throughout the story? What does the term "Black Man" mean and how is it used in the story?
To answer your other questions, the scarlet letter changes in meaning throughout the novel. At first it means adultery, but then as the novel progresses, some people in the town begin to think it means 'able' because Hester is known throughout the town as one who sits by the bed of the sick and helps the needy. She goes about these tasks silently without every asking for payment of thanks, and the townspeople eventually notice and some change their opinion of Hester. In one other part of the novel, when Dimmesdale, Hester, and Pearl are standing on the scaffold at night, a letter 'A' shows up in the sky from a meteor. Dimmesdale feels that is a symbol for his adultery, but many of the townspeople interpret it to mean 'angel' because the Governor had just passed away.
The term 'Black Man' is a term used by the Puritan people to mean the devil. 'Black' was referring to soot on his face from hell as well as an indication of evil. The term in the story is used to refer to Chillingworth after he begins to torture Dimmesdale. Little by little, as he seeks revenge, Chillingworth becomes more and more evil looking, and is described as looking like Satan himself.
Both Hester and Dimmesdale sin against the rules of the community when they fall in love and have sex--and they sin even more when that encounter produces a child, for that is the evidence of their sin, for the puritan community the biggest sin of all is "seeing" it, putting it in the midst of the community. Whether the author and narrator consider this a sin--that is another matter, for here is where the beautiful ambiguity of the story lies, Hawthorne's real mastery, for he plays this sin with such irony. As far as salvation, Hester appears saved in humanist terms from the beginning because she acknowledges her deed, pays for it with dignity, releases herself of guilt, and raises the extraordinary child named "Pearl." Dimmesdale is another story. He lies, and lies, and lies more by hiding his truth, and it is this hypocrisy that is his great sin that increases his guilt and distance from God, even though he remains "close" (ostensibly) to the community.