The Board of Aldermen of the town send a delegation to the home of Miss Emily Grierson because they want her to pay taxes. She does not pay any taxes on her home and property and they think that she should.
In a previous generation (in 1894), the mayor of the town had said that Miss Emily would not need to pay any taxes. But later, when a new generation came into power, they thought that this was not how things should be and they sent the delegation to ask her to pay (since she had ignored letters).
When Miss Emily Grierson's father died, it became clear to the townspeople that Miss Emily had no inheritance except the house she lived in. Colonel Sartoris, the mayor at that time, issued an edict that Miss Emily's property taxes would be forgiven for as long as she lived ("into perpetuity"). Colonel Sartoris made up a story about her father having lent money to the town and the town choosing to repay the debt in this manner. Miss Emily was informed, and thereafter she never paid her taxes.
As the old guard passed away, a new generation of town leaders arose. These men didn't like the fact that Miss Emily didn't pay her just share of taxes. They sent her tax notices and letters, which Miss Emily either ignored or returned without enclosing funds. The men decided that they would need to address the issue with Emily in person. The mayor sent a letter offering to come see her in person or to send a driver to pick her up. Miss Emily replied in writing that she never went out. At that point the town leaders sent a delegation of aldermen to her house. Presumably, although it is not stated, none of the men--including, it seems, the mayor--had the courage to take this duty on as a solo mission.
This hesitancy on the part of the aldermen to confront Emily in person shows that she still retained a certain mystique and status in the town. It also signifies her strength. A group of younger, stronger, smarter men is unable to overcome her stubbornness. In the end, "she vanquished them."