The characters mentioned in this poem are based on real life, historical figures. The narrator is Duke Alfonso II who ruled a place in northern Italy called Ferrara between 1559 and 1597. The Duchess of whom he speaks was his first wife, Lucrezia de' Medici who died in 1561 aged 17, only two years after he married her. In real life, Lucrezia died in suspicious circumstances and might have been poisoned.
The poem is set in 1564, three years after the death of the Duchess. An emissary (messenger or representative) has been sent to see the Duke from the Count of Tyrol. The Count is the father of the Duke's next wife (he married three times in all). The Duke shows the emissary a picture of his late wife and remarks on her character, suggesting that she was unfaithful to him - and hinting that he might have killed her because of it.
During his speech, the Duke makes himself look arrogant, insensitive and selfish.
"Looking as if she were alive. That’s my last Duchess painted on the wall"
"Fra Pandolf's hands Worked busily a day, and there she stands Wil't please you to sit and look at her?"
"Sir, twas not Her husband's presence only, called that spot of joy into the Duchess' cheek?"
"She like what'er she looked on, and her looks went everywhere."
"as if she ranked My gift of a nine-hundreds year old name"
"of mine for dowry will be disallowed; Though his fair daughter's self, as I avowed"