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Hester
The scarlet letter is meant to be a symbol of humiliation, subjugation, and a reminder of the canon of Puritanism. Since Hester was found guilty of being an adulteress, the magistrates feel that they had the right to single her out and render her vulnerable to the insults and judgement of the villagers, who are actually quite ruthless and heartless.
Yet, in their small mind mentality, the villagers grossly miscalculated a very important factor: Hester is an outsider who came to live in the village not that long ago when she was sent there from England by her husband. The latter never made it and was presumed to be dead. Therefore, for all we know, Hester may have come with a completely different set of thoughts and beliefs that distance her completely from the villagers of Boston.
We learn as readers that Hester's husband was taken prisoner by the natives, lived in the wilderness for a while, and then finally came to Boston under the adopted persona of Roger Chillingworth.
[Hester is] the wife of a certain learned man, [who] was minded to cross over and cast in his lot with us of the Massachusetts [so] he sent his wife before him, remaining himself to look after some necessary affairs.
And, from what we know, Hester has only been in Boston for about two years with no news of her husband.
This said, we have to be very careful and not assume that the scarlet letter meant "shame" for Hester. Hester has very little in common with the Bostonians. Otherwise, would a proper Puritan "goodwife" let her desires take a hold of herself to the point of getting pregnant from the saintly man of the village?
On the contrary, the scarlet letter, which meant to put Hester to shame, becomes Hester's very own symbol of individuality. Hester goes as far as designing her very own letter. She embellishes and decorates it to the point that it looked
... fantastically embroidered and illuminated upon her bosom. It had the effect of a spell, taking her out of the ordinary relations with humanity, and enclosing her in a sphere by herself.
Hester's scarlet letter is a declaration of her individuality, personal independence, and the marked desire to separate herself from a society that will never understand her.
Eventually, Hester works so hard on her own that she makes a name for herself as a seamstress. She creates beautiful garments for people, even for those who shun her and insulted her. The scarlet letter changes into different meanings for many people—some go as far as saying that the "A" stood for "able."
The letter was the symbol of her calling [...] many people refused to interpret the scarlet A by its original signification. They said that it meant Able, so strong was
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The letter was the symbol of her calling [...] many people refused to interpret the scarlet A by its original signification. They said that it meant Able, so strong wasHester Prynne, with a woman’s strength.
Therefore:
Hester Prynne is aware of what the villagers think about what wearing the scarlet letter will do to her. Yet, Hester sees no shame in her letter, even if she abides by the mandate to wear it as a symbol of shame.
According to the book Understanding the Scarlet Letter: A Student Casebook to Issues, Sources, and Historical Documents by Claudia Durst Johnson (1995), Pearl can be seen as the embodiment of the scarlet letter because she and the letter are the visual reminder of the "indiscretions" of Hester Prynne.
If you think about it, had Hester not become pregnant, she would have never been caught, and she would have never had to wear that pathetic letter on her bosom. Pearl seems to have come to burst the bubble of what once was a passionate, carefree love. She turned the fantasy into punishment. She is the cause of the scarlet letter. The text evidence of this can be found in chapter 7, titled "The Governor's Hall:"
[...] it was a remarkable attribute [...] the child's whole appearance, that it irresistibly and inevitably reminded the beholder of the token which Hester Prynne was doomed to wear upon her bosom. It was the scarlet letter in another form; the scarlet letter endowed with life!
Pearl is also obsessed with the scarlet letter. She teases her mother about it, asking her questions about what it really means. She tries to understand the origin of the letter no differently than she consistently asks about Dimmesdale, as if in aims to understand her very own origin as well.
Dimmesdale
The scarlet letter has a different meaning for Arthur Dimmesdale. He is actually the one who takes the scarlet letter as a symbol of shame. He is, literally, the only one wearing the letter out of pure shame and guilt. To make matters worse, he is wearing the letter in secret and in the worst way possible: carved on his chest. This is done as self-inflicted punishment since he feels the shame and guilt of not daring to tell his parishioners that it was he who impregnated the “adultress” Hester Prynne; that he has been a hypocrite to his people ever since he first dared to point a finger at her on the scaffold accusing her of the very “sin” that he helped her commit.
He punishes himself over and over for this, but the real punishment that he needs to endure is telling the truth. He is unable to do so and, therefore, he continues to suffer.
Chillingworth
Finally, the scarlet letter means a triumph for Roger Chillingworth. Witnessing Hester at the scaffold wearing the letter in shame emboldens him to carry out his revenge against both Hester and Dimmesdale. It is not easy to break Hester, except with threats. Eventually, he needs ultimate proof that Dimmesdale is directly responsible for what happened to Hester: proof to show that “the young divine” is everything but.
Posing as Arthur Dimmesdale’s personal healer under a false name, Roger Chillingworth discovers the carving on the priest’s chest, finally making the ultimate connection. Chillingworth uses the knowledge of the scarlet letter as his personal way to manipulate Dimmesdale into weakness.
Further Reading
For Hester, the letter is a daily reminder of her sin. While she does not seem to ever express regret about her actions, the knowledge of her sin does seem to cause her pain. However, it is also her visible link, along with Pearl, to her co-sinner, and for this reason, in part, she would never remove it. In addition, the letter is a daily reminder of her society's judgment. Although they ought to forgive, to exercise compassion, and to refrain from judgment according to their religion, they do the opposite for much of the novel. It sets Hester apart, no matter where she is, in her own sort of circle.
For Pearl, the letter is symbolic of her mother. She seems to think it is a good thing, as most little children think of things associated with their mothers, and Pearl wants her own letter, even constructing one of eel grass one day by the water. When her mother takes it off in the forest with Dimmesdale, Pearl refuses to listen or come to her until she pins it back on. Her mother is the only person who treats her kindly and lovingly, and so perhaps Pearl associates it only with those good things (as opposed to the Puritan elders and their children, who she so dislikes, and none of whom wear letters themselves).
For Chillingworth, the letter is symbolic of the wrong done to him by Hester's co-sinner. He has said that he and she are even—he wronged her by marrying her knowing that she could not love him, and she wronged him by being unfaithful—but there is one who has wronged them both. The scarlet letter, then, reminds him (as if he could forget) of the one person who he wishes to ruin by whatever means possible.
For Dimmesdale, the letter reminds him of his own secret sin. While Hester wears her shame openly, giving her the opportunity to atone for it here on Earth, Dimmesdale is conscious of the fact that he does not have such an opportunity because his sin is unknown and because he adds hypocrisy to sinfulness by keeping silent.
In order to fully answer the question, let us first assume that "scarlet letter" includes both the symbol of the punishment that Hester bears as well as the piece of cloth she wears. In this way, each of the main characters in the story are differently affected by the scarlet letter.
Hester Prynne: at first it is an embodiment of shame, isolation, and societal disgrace. However, as she accepts her fate, it comes to embody inner strength, empathy, and a burden she believes she deserves and eventually embraces.
Chillingworth: it is a daily reminder that he has been wronged (as Hester's husband) and his shame, though hidden, manifests itself in the form of rage and revenge. The letter, as Hester's punishment, allows him to forgive his once wife, in the belief that she has received what she deserves. But then it causes him to focus his rage and revenge on the other half of the letter: Dimmesdale.
Dimmesdale: because he does not physically wear a badge of shame like Hester, the letter symbolizes internal shame and torment, which may arguably be worse than what Hester must publicly endure. Seeing Hester punished while he hides his guilt drives him mad, makes him physically weak and sick, and the audience is lead to believe that he somehow comes to bear a physical mark of his shame on his chest as a result. He believes wearing the letter (admitting his sin) is the only thing that will free him from inner-torment.
Pearl: as an ironically intelligent and precocious child, the scarlet letter upon her mother's chest is a symbol of hypocrisy and public ignorance. She frequently laughs, plays, or childishly pokes fun at the letter itself, which reveals that her knowledge of society and its flaws go deeper than her age and experience should. In some ways, the letter causes Pearl to judge her mother, not because of what she has done nor her punishment, but for attempting to conform to the rules of a society which Pearl clearly sees are flawed. On the other hand, she does not recognize her mother without the letter, showing that it has come to such an innate part of her existence, that she cannot imagine life in its absence.