The resolution of a story comes after the climax. In "Sonny's Blues," the story reaches its climax as the narrator listens to and is moved by Sonny's music in a jazz club and suddenly understands and respects his brother for the first time.
Before this, the narrator had been troubled about this younger brother, a drug user who has spent time in prison. The narrator had felt concerned and responsible for Sonny going wrong. For some time, he pulled away from his brother, not contacting him until his own daughter died of polio. This narrator had spent years trying to "fix" Sonny.
But at the jazz club, listening to Sonny play the blues, the narrator gains new insight into and respect for Sonny as an artist who has gained the esteem of his colleagues and created art out of his pain in a way that can touch other lives.
As the narrator absorbs all this, he pays tribute to his brother. This is the resolution of the story: the narrator sends a drink, a scotch and milk, to the piano where his brother is taking a break from his playing. It is a sign of respect and approval. At first, the brother doesn't know if Sonny has truly noticed or understood it, but when he "sip[s] from it and look[s] toward [him], and nodded," the narrator knows the gesture has been understood and accepted. The two men have reached a reconciliation.