The speaker points out a lifelike portrait of his "last Duchess" that’s painted on the wall.
This tells us that the speaker is a Duke, that his wife is dead, and that someone is listening to him describe his late wife’s portrait, possibly in his private art gallery.
It also makes us wonder what makes her his "last" Duchess – for more thoughts on that phrase, check out our comments in the "What’s Up With the Title?" section.