The narrator of the poem, a duke, disliked that his last duchess had "A heart [...] too soon made glad, / Too easily impressed." He wanted it to be his presence only that would call "that spot / Of joy" into her cheek, but, instead, the duchess seemed to be equally thankful for everyone and everything. People would bring her cherries from the orchard or a white mule to ride on, or she would see a beautiful sunset, and she counted those things as equal to his "favour at her breast." She displayed the same amount of gratitude for gifts both small and large, as though "she ranked / [His] gift of a nine-hundred-years-old name / With anybody's gift."
He wanted her to value him and the things that he gave her, like his name, above anything and everything else; he wanted to have more of her smiles and blushes than anyone else. However, she was simply a happy person who seemed to take joy in even the smallest of tokens.
The duke admits that he could, theoretically, have told her about his preference, or explained that he felt she did not value him highly enough, but—to him—this would constitute "stooping; and [he] choose[s] / Never to stoop." In other words, he thinks it is beneath him to have to explain to someone why they ought to value him the most. So he "gave commands; / Then all smiles stopped together." It sounds as though he's had her killed. Now he can start over with a new duchess, it seems.