dbello's answer covers most of the ground, but I have another view of her last comment, that her "sin" was her freedom. Hester was "free" before her sin; she had an open attitude toward life that allowed her to see her failing as a violation of civil law but not a violation of her internal law. After her sin, she was clearly less free; there were places she could not go, she had, at least temporarily, lost the respect of her fellow citizens, she had to raise Pearl by herself (no easy task), and she was destined to live alone.
She could bear all this because of her integrity and strength, but these were there before her "sin."
Hester Prynne definitely regards herself as an individual not afraid to utilize her own thoughts for herself. She demonstrates personal strength, a quality the two male characters Dimmesdale and Chillingworth do not possess. Hester is beautiful, young and has an independent spirit. This is in great contrast to the Puritan community she lives in. The best example of how Hester identifies herself as a woman is her belief in her own humanity. True, she broke a Commandment and for that was to be punished. She accepts punishment with grace, however never relinquishs her passion for her lover, attachment to her child, nor her disdain for the Governor. Hester held true to her beliefs, and as a result the experience allowed her strength to grow stronger. The effect of the "sin" was her freedom.