Mikhailov assumes that Vronsky and Anna are wealthy Russian aristocrats who like to pretend to know a lot about art.
Mikhailov first shows them a scene called Pilate's Admonition.
(Some background information: Pontius Pilate, governor of Judea during the life of Jesus. He weakly allowed Jesus' powerful enemies to press him for Jesus' conviction. Following his crucifixion [in the Eastern Orthodox account of Pilate's life] Pilate and his wife convert to Christianity, and she is considered a saint. For more on Pilate, check out this article on his historical and religious roles, and see also our theme on "Compassion and Forgiveness.")
Golenishchev compliments Pilate's facial expression, saying that he seems like "a kind, nice fellow, but a functionary to the bottom of his soul" (5.11.8). Mikhailov is delighted: this is exactly how he imagines Pilate, as a bureaucrat who's ignorant of what it means when he signs Jesus' death warrant.
Mikhailov loves Golenishchev for seeing what he meant to express in his painting.
Then, Anna, perceiving that Christ is supposed to be the center of the painting, compliments the pity in Christ's face. Once again, Mikhailov is in ecstasy, because that's exactly what he wanted to show: Pilate's bureaucratic soul represents physical life, while Christ's pity shows his spiritual depth. Mikhailov is getting symbolic with this painting, because he doesn't believe in God. He's trying to paint both Christ and Pilate as types of men.
And then Vronsky has to spoil it all: he comments on the technical aspect of Mikhailov's painting, and now all that Mikhailov can think is that his painting falls short technically of what it's supposed to be like. He much preferred Anna and Golenishchev's more abstract comments.
Mikhailov and Golenishchev fall into an argument about Mikhailov's depiction of the historical Jesus, and Anna and Vronsky start talking to each other.