Symbolism, Imagery, Allegory
Jazz music symbolizes different things to different characters in this story.
The narrator doesn't know anything about jazz. He associates it with a certain "element" of people, people he doesn't want his brother hanging out with. He lumps jazz together with drugs and Sonny's addiction, blaming the jazz lifestyle for turning Sonny into a heroin addict because he knows that some musicians have to get high in order to play. Jazz music makes the narrator angry and bitter.
But for Sonny, jazz music is like a ray of light. He loves playing it and listening to it. It's the one really positive thing in his life. Jazz music represents passion and escape for Sonny. The very people the narrator negatively associates with jazz are the ones who function as a sort of second family for Sonny. While jazz is alien to the narrator, it's comfortable and comforting for Sonny.
At the end of the story, jazz functions as a bridge between the two brothers. When the narrator goes to see Sonny play, he learns something about his brother that he's never understood before. When he hears Sonny play, he finally starts to appreciate the wonder and terror of being a musician.