There are several conflicts in James Baldwin's "Sonny's Blues."

One of the main conflicts in the story is between the narrator and his brother Sonny: man vs. man. The narrator has worked hard to become educated and find a job that can support him and his family. Sonny, however, is haunted—he cannot survive as his brother does. He has taken a more difficult road, in that supporting oneself as a musician is competitive, and work is often sporadic. 

Sonny isn't very inclined to share his inner feelings, another aspect of his personality that keeps the brothers distant:

Daddy was big and rough and loud-talking, just the opposite of Sonny, but they both had—that same privacy.

Sonny is also an addict, and this too creates conflict with Sonny's brother because he does not know how to connect with Sonny, as he does not understand the draw of addiction. The narrator also feels responsible for Sonny because of a promise his mother exacted from him before she died: to watch out for Sonny, who is seven years his junior.

The other conflict that most adversely affects both brothers is man vs. society. Each one, as an African American in society in the 1950s, exists during a time when white men dominate seemingly every aspect of American life. And each brother must struggle to put his past of living in the poorest of the New York neighborhoods (Harlem) behind him, each attempting to find his way in the world. The narrator has become a teacher. Sonny struggles to be a musician.

There is darkness in each of their lives, and the author intimates that it is something neither can avoid. The narrator talks about what waits for kids (such as they were) as they grow up. Sonny's brother remembers sitting around with the older folks on Sunday afternoons as they considered the world around them. The lowering sun brought darkness into the room, and there was comfort with the older folks being there. Then someone turns on the light, the grownups stop speaking as they had earlier, and the child senses a change:

And when the lights fill the room, the child is filled with darkness. He knows that every time this happens he's moved a little closer to that darkness outside. The darkness outside is what the old folks have been talking about. It's what they've come from. It's what they endure.

The narrator recalls somehow knowing that what happened to the older folks would soon happen to him. It was not something for which they could prepare their children, except to keep the knowledge from them for as long as possible.

This, then, is the stark reality of man vs. society for these young black men moving away from their childhood. While the narrator conforms to society's expectations of becoming a functioning, acceptable part of the social machine, Sonny seems unable to do so. The darkness seems to have crept into his heart, and the drugs may serve to hold it at bay. Sonny says:

[Heroin] makes you feel—in control. Sometimes you've go to have that feeling.

The only time Sonny experiences freedom is when he plays his music: when the world seems somehow balanced and more tolerable. It is at these moments when Sonny comes alive, when he "can stand it."

When Sonny begins to play, his music speaks first of grieving. The notes represent his life. Then suddenly the music shifts and it is no longer a "lament". Somehow, it rises above the conflict society represents:

Freedom lurked around us and I understood, at last, that he could help us to be free if we would listen, that he would never be free until we did.

The music holds the "hungry tiger" of society, which waits to devour them, at a distance for as long as the music lasts. This, then—and the drugs—allows Sonny to deal with this social struggle. It is only at the story's end that the narrator recognizes this conflict and the divergent ways in which each brother fights to stand against it.

There are many conflicts in this story, both external and internal.  A few of the internal ones are in Sonny and his brother, the narrator.  Sonny's internal conflict is about his battle with drugs and his creativity.  He seems to believe, rightfully or not, that he cannot be musically productive without the drugs.  One of the narrator's internal conflicts is his love for and his distaste for his brother.  There is, of course, an external conflict between the narrator and Sonny, each of whom has chosen very different paths in life.  There are also societal conflicts, which are those between the two different races.  There is a constant use of lightness and darkness, throughout the story, which symbolizes these conflicts.