Faulkner's short stories, like his novels, contain many themes, and one of the most consistent is the struggle between what we will call the "Old South" (the pre-Civil War south) and the "New South," which grows out of the Civil War and the Reconstruction period. In this story, Miss Emily is part of the Old South, as is her father and Colonel Sartoris, and she is constantly in conflict with the rising New South.
For example, the New South, represented by the town council, tries to get her to pay city taxes in Jefferson. She refuses on the basis that Colonel Sartoris remitted her taxes permanently because her father loaned the town some money, and the town is paying the loan back by remitting Miss Emily's taxes. Faulkner tells us that Colonel Sartoris made up the story in order to, in effect, provide some charity to Miss Emily, which she would not have accepted had she known it was charity. As she does in other battles with the town, Emily prevails and sends the town council packing.
Perhaps more important, however, to your question is a comment Faulkner made about why he named the story "A Rose for Emily." Essentially, he said that Emily deserves a rose because she had what Faulkner called a "blighted life," a life ruined by her cruel and repressive father, who kept her from having a normal life by not accepting any of her suitors when she was young. Faulkner pointed out that if someone is repressed and denied the opportunity to have a normal life--with a husband, a home, children--that repression may come out in horrible ways.
The story centers on repression and how it can twist a soul into something unrecognizable.
Faulkner's short story "A Rose for Emily" serves as a warning to his reader. In it, we learn that nothing, no matter how much we might wish it to, truly lasts forever. Emily tried to keep things around her the same, but she soon learned that she could not prevent change from occurring. When her father dies, Emily refuses to acknowledge it and refuses to let people in to remove his body for three days. Later, while seeking to keep Homer, her first real beau, from changing, growing, and leaving her, she poisons him with arsenic and sets his corpse in their bed. Rumors had spread through the town that he wasn't willing to settle down, but she prevents him from moving on by killing him and keeping him as hers.