In The Scarlet Letter, what business did Hester have at Governor Winthrop's house in Chapter 12?
Hester was "watching at a death-bed" as the onetime governor's health was failing. Members of the community would often go to a home when someone was about to die in order to offer comfort and support to the family, as well as to say goodbye to the ailing person. We learn, in the next chapter, that Hester has made herself quite useful in this way. The narrator says that when a household was "darkened by trouble," Hester came, and "there glimmered the embroidered letter, with comfort in its unearthly ray. Elsewhere the token of sin, it was the taper of the sick-chamber." In fact, Hester's presence was so common when anyone in the town was suffering—from poverty, from illness, or the death of a loved one—that people began to say her scarlet "A" actually stood for "Able," rather than "Adultery." Hester began to be known not for her sin any longer, but for "her many good deed since." Townsfolk would actually claim her as "our Hester—the town's own Hester" because she had become such a beacon of humility and helpfulness.
Governor Winthrop passed away that evening, and Hester tells Dimmesdale: "I have been watching at...Governor Winthrop's death-bed, and have taken his measure for a robe." So, she had been offering comfort and support during the Governor's illness and passing, and after he died, she took his measurements so that she could go home and make him a robe for his burial. Her good works for much of the book were in play even this evening in play. Dimmesdale is at the scaffold, tormenting himself as usual, and he speaks with Hester and Pearl as they are headed home. I put a link to a more complete chapter summary below. I hope this helps!
Hester has been sitting at Governor Winthrop's deathbed, sewing a burial shroud for him. When she and Pearl are returning home late at night, they find Dimmesdale standing on the scaffold. It is at this time that the mysterious scarlet letter appears in the dark sky.