This is such a complicated but wonderful poem. The Duke of Ferrara is a very selfish man who considers everything within his grasp "his". He owns many things...the bronze statue, for instance. The most important thing he "owns," however, is his last duchess. She hangs on the wall as if she were still alive--no one sees her unless he approves, otherwise she is hidden behind the curtain for his eyes only.
The Duke is talking the entire time, and he reveals much of his selfish and haughty personality through what he says...and what he does not. Read between the lines. He tells the envoy of the New Duchess that he will not tolerate someone who does not live for him entirely. So warn your man that his daughter better suck up to me or she'll end up on the wall like my last duchess.
The Duchess found pleasure in simple things. She did not respect his gift of a very old and distinguished name, and she smiled at everyone...not just the Duke. She appreciated it whend someone brought her flowers or fruit, or showed her any kindness whatsoever. So, rather than lower himself to tell her that he didn't like this, he gave orders to "stop her smiles" altogether. We understand that he had her killed.
Some critics read into this that the Duchess was cheating on the Duke, but I think she just enjoyed simple pleasures of life--sunsets and friendly people and juicy peaches, etc.