Simply, the letter A on Hester's gown brands her as an adulteress for all the world to see. Her sin/crime has been exposed, and wherever she goes, people will know what she has done. It also serves to brand Pearl as the product of adultery.
The significance of the letter "A" on Hester's gown stands for adultery. Hester was convicted of adultery , and sentenced to wear the symbol . Later in the novel, it takes on a new meaning "able" for all the good, charitable work Hester does.
The A on Hester's chest stans for Adulterer. It is her punishment for having Pearl. It defines her as the outcast from puritan society and lets her sin be known to all. Later as the community comes to accept Hester again near the end of the novel, the A is believed to stand for Angel. She is once again accepted into society and forgiven.
A (which stands for adultery) is a physical reminder of her "sin".
The simple answer is that the "A" simply stands for adultary. However, I believe it stands for the conflict within the story between Puritan society (rigid/does not allow for emotional flexibility) and Hester's natural beauty and dignity.
The "A" is supposed to have a negative stigma attached to it. However, many of the same people who believe this see the "A" formed by the meteor as meaning Angel. The two A's are not only the same letter but similar in color.
The contradictions in the society and in Hester herself continue through the book and are exemplified in this letter "A".