One of the the main conflicts with Hester is that she has to live with a badge of shame while she, in fact, does not feel at all shameful. She has to struggle against a population of people who act in a "holier than thou" manner and use her as a scapegoat for their own prejudices and as a filter of their own weaknesess and hatred.
However, internally, there is very little evidence in the story that shows Hester's true feelings either about what she did nor for Dimmesdale. Could it be that Hester simply did not care about committing the indiscretion? Could it be that Hester, from the get go, could not have cared less about what she did and that the only thing that came in the way her and Dimmesdale's passionate personalities was her untimely pregnancy.
Therefore, Hester's inner conflict is having to fight a hypocritical society that feeds off her mistake to sanctify itself.
I think the easiest way to answer this question is to consider how you would feel if you were Hester Prynne. In order to do this, you must consider the lengthy list of circumstantial conflicts she experiences:
- Married to a man she does not love and arrives at the New World without him.
- Falls in love with a man she cannot have and has a baby with him.
- Arguably unfairly punished for her "sin" by a Puritan society who never relents in reminding her of it.
- So devoted to her faith/religion that believes she deserves her punishment, but devoted enough to the father of the baby that she will not reveal his secret (and therefore let him bear part of her punishment).
Considering these things, understand the emotional conflict that Hester must have raging inside her. She feels truly guilty and is genuinely repentant for what she has done. This is obvious by the way she chooses to live, how she chooses to raise her child, and the fact that she wears the letter until the day she dies.
On the other hand, she is still madly in love with the father of Pearl. This is evident by the fact that she will not, under any circumstance, give up his identity.
Putting all of this together, consider how you might feel if the one person you loved most in the world, you could not have, and you actually agreed with the (in this case, religious) reasons that prevented you from having him? This is exactly the internal conflict Hester experiences.