Browning's poem is an excellent example of a dramatic monologue. It is a monologue because the entire poem consists of the words spoken by a single voice, Duke Ferrara's. It is dramatic because it contains the elements of drama, primarily characters, a narrative, physical action, and a physical setting. Through these devices of drama, the poem becomes a scene that could be staged with actors and props.
In the poem, the Duke speaks to another person, perhaps an emissary from the family of his next wife. The listener does not interrupt, but his presence is felt in the poem as the Duke refers to him as "you" and "sir" and offers him a chair so that he may sit and look at the portrait of the Duke's former wife. At the conclusion of his comments, the Duke directs his visitor to leave his seat and move on:
Nay, we'll go
Together down, sir.
As they descend, the Duke points out a bronze statue of Neptune taming a sea horse, symbolic of his relationship with his former wife.
These physical actions (sitting and leaving) and the props (the guest's chair, the portrait of the Duchess, and the bronze statue) make it possible for the reader to visualize the scene as if it were played out upon a stage.
Throughout the Duke's monologue, the story develops, and his character, as well as the Duchess's, is revealed. Her fate at his hands is not established precisely, but the conclusion of the poem (and the story) implies that his next Duchess will be marrying a man who misrepresents himself, concealing the arrogance and cruelty of his nature.