As you look for evidence of the Purtian society in The Scarlet Letter, look at Hester herself. She experiences a guilt that only a society of Puritans can put on her. She makes comments about how even when the society is ready for her to lose the letter A, she won't take it off until it naturally falls off of its own accord. (This occurs in between 14 and 16 I believe) This is the life of a Puritan, never feeling "good enough" for God and always working toward a purity that is perfection which as humans we know is unattainable. The problem with this guilt is that the bible offers forgiveness too. Puritans never seemed to accept that entity of faith.
The community gave Hester a terrible time, never letting her into social circles because of her great sin and they maintained a terrible judgement of her daughter Pearl. The feeling of outcast never quite left Hester through the entire work. Even though she had a great skill to offer and had finished her time, their social punishment lingered. They could not concieve of forgiveness in their own right and therefore did not offer it.
Much of Hawthorne's writings include direct or implied criticism of Puritanism - a system that punished what it considered to be sin or sinners over-harshly and seemed to breed hypocrisy. Here we have many moments of such hypocrisy, mostly based in the character of Arthur Dimmesdale. For me, the most ironic moment comes when it is Arthur Dimmesdale, in his role as clergyman, must remonstrate with Hester Prynne to divulge the name of the person she committed the sin of fornication with, when all along he and Hester knows that it was him.
Interestingly though, overall the structure of this novel registers historical change as we move back from Massachusetts in the nineteenth century to Massachusetts in the seventeenth century. The book identifies a historical trend towards greater liberality, and that is broadly identified with the softening of social attitudes. Hawthorne couches this in terms of gender, suggesting that the conventionally masculine attributes which dominated Puritan Boston have been displaced by more conventionally feminine qualities, including compassion, sensitivity and aesthetic awareness. There is room, however, for further development, as is made clear through the ending of the novel.