The Scarlet Letter opens with the main character, Hester Prynne, imprisoned and then shamefully subjected to public scrutiny and humiliation on the scaffold. At this point, the reader has no inkling that Prynne will somehow turn her personal tragedy into a positive thing. Author Nathaniel Hawthorne then devotes the next several chapters to introducing and developing another important character, Roger Chillingworth, who will play a critical part in the story as the plot unfolds. It is not until chapter 5 that Hawthorne focuses solely on the character of Hester Prynne and the effect of her ongoing punishment, the scarlet letter, on her life.
By chapter 5 it is already known that Prynne has been outcast as an adulteress, but in this chapter the reader sees specifically how this has affected her, as in this line:
"Continually, and in a thousand other ways, did she feel the innumerable throbs of anguish that had been so cunningly contrived for her by the undying, the ever-active sentence of the Puritan tribunal. Clergymen paused in the street to address words of exhortation that brought a crowd, with its mingled grin and frown around the poor sinful woman."
Then, later in the chapter, Hawthorne adds depth to Hester’s character by showing that the scarlet letter has given her a new ability, something that she did not possess before her suffering began:
". . . she felt or fancied, then, that the scarlet letter had endowed her with a new sense. She shuddered to believe, yet could not help believing, that it gave her a sympathetic knowledge of the hidden sin in other hearts."
Hester is now able to empathize with others in a way that she couldn’t before. The fact that she can do so when she has suffered so much reflects very positively on Prynne’s character. This sets the scene for information that the reader gets much later in the story, when we find that the townspeople have gradually put aside their disapproval of her and now actually perceive that the “A” she wears on her breast stands for “able.”
You can find this answer in the chapter about Hester's life alone, outside of the town. I believe this is 7 years or so after Pearl's birth. We see that the letter comes to mean "Able" and not "Adulturer," as originally intended. She is able to live as she pleases, making a life for herself and her daughter. Unheard of for a woman at the time! People come to her for her embroidery, and so to some extent include her in society. Yet, she still lives separate from the town, and is not subject to the rules and limitations that they are.