In this poem, the duke explains what happened to his last duchess, his former wife, saying that she was "too soon made glad," and so he "gave commands" which stopped all of her smiles. We know that she is dead now, and that the duke is seeking a new wife, and so we can infer that the "commands" the duke gave had to do with disposing of his last wife. He had her killed so that he could marry again, so that he could find a wife that would not rank his "gift of a nine-hundred-years-old name / With anybody's gift."
See, the trouble with his last duchess is that everything made her equally as happy: the sunset, a bough of cherries, a white mule. She would thank everyone as though they had given her something as significant as what her husband had given her—his name—and this made him jealous and wounded his pride. He feels that to have had to explain to her why she should smile most for him, to blush only for him, would have amounted to "stooping," to lowering himself. The duke was simply unwilling to explain, and he wants a wife who will love him the best, who will be most impressed by him, and so he got rid of his last wife to make room for the next.
The fact that he explains this so coolly, as though it were a perfectly reasonable course of action, is one of the more chilling aspects of the poem.