The A in The Scarlet Letter changes meaning the as the town's view of Hester changes. At first, the letter means Adultery. As Hester takes the scaffold in front of the town, the magistrates explain her sentence is to wear the A as a symbol of her sin of adultery.
The second meaning of the A is when the town sees the meteor in the sky the night of Dimmesdale's vigil. While the reverend sees the A as nature mirroring his guilt, the towns people who see the A in the sky believe that it means Angel ushering Governor Winthrop's soul to heaven.
The final meaning of the A is able. As the town's view of Hester changes the stop thinking of her A as adultery but as able. She sews for the people in town (for everything except weddings) and comforts the sick. Women who are heart broken, or people who need a comforting ear found solace with her. They believe the letter has done its job and transformed their Hester.