The title of "My Last Duchess," like the first few lines of the poem, gives us quite a bit of information about the dramatic scenario in the text. The word "My" clues us in to the fact that the poem is going to be in the first-person – so, before the poem even begins, we know from the title that we’re going to be hearing the voice of a character, not just of a general poetic speaker. The title "Duchess" makes it clear that we’re dealing with European nobility, probably in a bygone age. And then there’s that adjective "last."
We’ll go ahead and ask the obvious question for you: why is she the Duke’s last Duchess? Well, that implies that there will be another Duchess in the future – and that there might have been several Duchesses before her. But wait a minute, isn’t marriage forever? Not for this Duke, who seems to dispose of Duchesses pretty quickly. So the designation "last" implies that this Duchess is only one of a sequence, preparing us for the fact that the poem might consider some of the other women who end up in that sequence.
After all, when you start describing something as your "last" one, you’re usually about to start explaining what’s going to be different about your "next" one, as in "My last car always hydroplaned, but I’m going to make sure my next one has good traction." The Duke’s last Duchess smiled and showed favors to everyone, and the Duke is going to make sure that his next wife, the daughter of the Count, doesn’t behave in the same way. How is he going to do that? Well, telling this story to her father’s servant, and thereby warning everyone in the Count’s household that he murders wives who are nice to their servants and male friends, is probably a pretty good start.