This question encompasses the entire novel and cannot really be answered in the detail you want in this short space. I suggest you consult the links below for more information.
1. Hester changes from a woman who flaunts her sin by wearing a beautiful dress and skillfully embroidering the letter "A" to a woman who is dressed more conservatively and is stronger both emotionally and physically than she is at the beginning of the novel. This is because she her sin is open and she learns from her mistakes.
2. Dimmesdale becomes weaker and weaker as the novel continues. He agonizes over his hidden sin, to the point that he puts his own secret letter "A" on his chest. He tries to confess publicly, but stops his confession many times until the end of the novel. The stress of the public confession is too much for his heart and he dies.
3. Chillingworth becomes totally focused on revenge during the novel. This has an effect on him physically as he becomes more and more deformed. Hawthorne says he was initially a fairly kind healer, but he becomes almost devilish in his search for revenge. After Dimmesdale dies, he has nothing more to live for and he withers up like the leech he has been described as, and dies.
To add on to the previous response:
1. Hester begins the novel almost immediately. In the second chapter of the novel, she is talked about by the women of the town as a "hussy," and only one of the gossips encourages the other women to consider that all are sinners; Hester just happened to be caught. When one is immediately seen as such a pariah, one naturally acts as such. Yet Hester, though she boldly embroiders the scarlet letter on her bosom, does not actually act that bold throughout the novel. Instead, she takes on her sin and attempts throughout her life to atone for it. In chapter 5, Hawthorne comments on the fact that Hester could have fled town and returned to England, but instead, she stayed (although at the edge of town) and used her needle-work skills to make a living:
Her needle-work was seen on the ruff of the Governor; military men wore it on their scarfs, and the minister on his band; it decked the baby’s little cap; it was shut up, to be mildewed and moulder away, in the coffins of the dead. But it is not recorded that, in a single instance, her skill was called in aid to embroider the white veil which was to cover the pure blushes of a bride. The exception indicated the ever relentless vigor with which society frowned upon her sin. (Chapter 5)
Thus, socially, Hester was eventually accepted, and while at first the memory of her sin would stay with the townspeople, that memory, too, would fade. By the end of the novel, the "A" that was emblazoned on Hester's chest came to stand for "angel" instead of "adultery" because of the very acts that Hester showed to those in the town.
2. Spiritually, Dimmesdale's change is interesting to note from Hawthorne's perspective. Because Hester was able to confess her sin (even though she did not want to), she was able to unload the guilt associated with it. Dimmesdale, on the other hand, kept his secret to himself, which is why he physically harmed himself, carving the "A" into his chest. When Dimmesdale finally confesses his sin to the congregation, the release of that sin causes him to die. Had he confessed during the multiple opportunities he had in the novel, he may have been able to live a full life.
3. With Chillingworth, again Hawthorne's commentary is interesting to note because it focuses on the use of magic and other dark arts. Throughout the novel, Chillingworth is viewed as a doctor, but he does not follow the traditional medicinal protocol. He mixes Native American herbal remedies with other scientific concoctions. Because this novel portrays a Puritan society, who would have viewed these types of remedies as black magic, Hawthorne shows the hypocrisy of such views. Chillingworth was a good doctor, but at the end of the novel, Hawthorne writes, "Others contended that the stigma had not been produced until a long time subsequent, when old Roger Chillingworth, being a potent necromancer, had caused it to appear, through the agency of magic and poisonous drugs" (Chapter 24). The town had viewed Chillingworth as a saint until the point when Dimmesdale revealed his sin and collapsed. That act ruined Chillingworth as well.
Hawthorne uses these three characters to reveal his view on sin and guilt in Puritan society and in his own society. While humans want to place the sin on others, they do not realize their own sin and its effects on their lives.