Emily Grierson is a symbol of the Old South and its refusal to die in the face of modernity. The townspeople certainly consider Emily to be as much. When she dies, many come to her funeral, because they see her as a "fallen monument" to the South as it was before the Civil War and Reconstruction changed it.
Emily fights the modern world constantly. She refuses to pay taxes because she claims Colonel Satoris, the old mayor, made her exempt from them. She refuses to allow numbers to be put on her mailbox. She still keeps a Black house servant in a relationship reminiscent of that between masters and slaves in the Old South. Most of all, she never mingles with the townspeople and eventually shuts herself up in her own home, further isolating herself from the modern world.
Despite what power she has, Emily is also a victim of the Old South's values, as evidenced by her submissive role in her relationship to her tyrannical father. He prevented her from marrying anyone and thus leaving him when he was alive. Her aristocratic upbringing has made her cold and unable to form relationships with those she deems lower—with the exception of the rakish Homer Barron, a Northern day worker who becomes her lover. However, even in that relationship, her entitlement and isolation inspire her to kill him when it becomes clear that he will not marry her.
Ultimately, Emily is a symbol of the Old South's unwillingness to fade away. Everything about her behavior, from her refusal to pay taxes to her keeping Homer's corpse locked away in her house, reflects her symbolic role as a "monument" to that way of life.