The main man vs. man conflict in The Outsiders is between the Greasers and the Socs, and specifically between Ponyboy and Bob in the fight in the park.
In literature, conflict is a struggle between opposing forces. The main categories of conflicts are internal and external. An internal conflict is an inner struggle, or a struggle against oneself (such as a decision or a fear). An external conflict, on the other hand, is a conflict between a character and an outside force. A man vs. man (or character vs. character) conflict is a struggle between two characters. There are also man vs. society, man vs. supernatural/magic, or man vs. nature conflicts.
The two gangs, the Greasers and the Socs, are at odds with one another over turf. The Greasers are working class kids and the Socs are socialists, or rich kids. They fight because they always have. They fight because of class differences, to settle beefs, and for control of the neighborhood. Specifcally, Ponyboy gets attacked by Bob and his friend David during the fight in the park.
Bob shook his head, smiling slowly. "You could use a bath, greaser. And a good working over. And we've got all night to do it. Give the kid a bath, David." (p. 55)
This external man vs. man conflict is actually the driving force of the plot, because Johnny and Ponyboy have to flee after Johnny kills the boy in the park.
The man vs man conflict would be Johnny and Pony vs Socs.