According to the article on "Historical Context" in the eNotes Study Guide:
The incident the poem dramatizes comes from the life of Alfonso II, a nobleman of Spanish origin who was Duke of Ferrara in Italy during the sixteenth century. Alfonso's first wife was Lucrezia, a member of the Italian Borgia family and the daughter of a man who later became pope. Although she died only three years into the marriage—to be replaced, as the poem suggests, by the daughter of the Count of Tyrol—Lucrezia transformed the court of Ferrara into a gathering place for Renaissance artists, including the famous Venetian painter Titian. (See reference link below.)
"My Last Duchess" is Robert Browning's most often anthologized dramatic monologue. As in all his dramatic monologues, there is a single speaker, and in "My Last Duchess" the speaker is the Duke of Ferrara. One of the features of Browning's dramatic monologues is that the speaker unwittingly reveals a great deal about his own character while addressing some other person or persons. In "My Last Duchess" the Duke is talking to a man who is representing the father of the young girl the Duke intends to make his "next duchess." They have come upstairs to discuss the matter of dowry, and the Duke is showing the visitor a portrait of his former wife, a beautiful young woman whom he apparently had killed because she did not please him.
Towards the end of the poem the visitor apparently jumps out of his chair and starts to leave the room without a word of thanks or goodbye and without having settled the matter of the dowry. It would appear that the visitor is so revolted and horrified by what he has heard that he intends to advise his master not to consider allowing his daughter to marry this greedy sadist. The Duke, who does not realize how much he has revealed about his character in his monologue, hurries after the departing visitor with the following words:
Will't please you rise? We'll meet
The company below, then, I repeat,
The Count your master's known munificence
Is ample warrant that no just pretence
Of mine for dowry will be disallowed
Though his fair daughter's self, as I avowed
At starting, is my object. Nay, we'll go
Together down, sir. Notice Neptune, though,
Taming a sea-horse, thought a rarity,
Which Claus of Innsbruck cast in bronze for me!
Notice how these lines, like all the lines in the poem, form rhyming couplets; but the couplets are intentionally made ragged and ungainly in order to further characterize the Duke as a pretentious lout who has no aesthetic taste although he obviously thinks very highly of himself as a connoisseur. The Duke himself admits he is not especially articulate:
Who'd stoop to blame
This sort of trifling? Even had you skill
In speech--(which I have not)--to make your will
Quite clear to such an one, and say, "Just this
Or this in you disgusts me; here you miss
Or there exceed the mark"
From the Duke's awkward description of his last duchess, it is hard for the reader to see any fault at all in the lovely young woman. It becomes apparent to his visitor that what his host is trying to say is that he disapproved of her because she was everything he is not. She was an excellent example of humanity, whereas he is an ugly, selfish, greedy, insensitive villain. Hopefully, the visitor is hastening to warn the Count, who is evidently being entertained downstairs along with his wife and other members of his retinue, against sacrificing his daughter to this ignoble nobleman.