The dead body is that of Homer Baron, the fiance, that the townspeople thought had abandoned Miss Emily. She poisoned him with the arsenic that she bought from the pharmacist, and then she kept his body in the house. That would be the smell that the town's fathers dealt with when they snuck around in the dark sprinkling lime around the house. The presence of a long, gray hair indicated that Miss Emily had slept, for quite some time at least, with the body of her dead fiance.
it is the body of homer barron. he isn't emily's fiance. they were not engaged. emily poisoned him because she doesn't wan't homer to go. homer isn't a marrying man as said in the story, thus when the construction of the sidewalk is done, homer has to go and leave emily but emily did not want that to happen so she killed homer barron.