In these lines, the Duke is showing his guest a painting of his last Duchess. He invites his guest to sit and look at the painting and points out the earnest and passionate expression on the Duchess's face. The Duke also mentions that there is usually a curtain in front of the painting, which only he is allowed to draw aside.
In the second line, the Duke says that in the painting the Duchess looks "as if she were alive." This implies that she is no longer alive. The Duke also, in the third line, comments on how accomplished the painting is. He calls it "a wonder," and says that the artist's hands "Worked busily a day" to paint the picture. There could be a double meaning to this reference to the artist's hands. The Duke could also be implying that the artist was too friendly with the Duchess and touched her inappropriately. This implication fits with the Duke's complaints elsewhere in the poem that the Duchess was too friendly with other men.
In the eighth line, the Duke points out the "depth and passion" of the Duchess's "earnest glance." When she was alive, it was this "passion" that caused the Duke to be jealous of the Duchess. He didn't like it that the Duchess was, at least as far as he was concerned, overly friendly with other men, blushing and smiling for everyone she met. Later in the poem he boasts that he "gave commands; Then all smiles stopped together."
The reference to the curtain in the tenth line is important because it alludes to the power that the Duke has over the Duchess now that she is dead. He is able to control who sees her and the story people hear about her. The Duke enjoys having this power over the Duchess now that she is dead, in part because he never was able to fully control her when she was alive.