There is no specific conflict with Johnny. Johnny is targeted because he is a greaser. He kills Bob in self-defense.
The conflict between the greasers and the Socs is a socioeconomic one. They are fighting because they are from two different social classes. The Socs are rich and the greasers are poor. They are also fighting because they are stuck in a cycle of violence. One fight begets more, because each party feels it needs to get revenge due to the last fight.
The greasers and Socs are street gangs of kids and young adults. Most of the criminal activity is not very serious, but some of it can get extremely violent and even lead to murder. These are not gun fights, normally, although guns are available in Ponyboy’s world. They are knife and fist-fights.
The most common altercation between the two gangs appears to be “jumping” followed by rumbles. Rumbles are planned, large groups of members of each gang meeting to fight it out. Jumping is an attack where a few members of one gang blindside one member of another gang.
Greasers can't walk alone too much or they'll get jumped, or someone will come by and scream "Greaser!" at them, which doesn't make you feel too hot, if you know what I mean. We get jumped by the Socs. (Ch. 1)
Jumping is violent and cowardly, because the person getting jumped really can’t fight back. He is outnumbered. The more severe weapon he has, the more dangerous the situation is for him. All he can do is let the other guys beat him up.
A rumble is also a serious situation. It is an agreed-upon meeting of the two gangs, with weapons (usually knives but not guns).
A rumble, when it's called, is usually born of a grudge fight, and the opponents just happen to bring their friends along. (Ch. 1)
Two-bit declares that a rumble is not a “fair fight” (Ch. 2). Weapons brought to rumbles, in addition to fists and knives, include “chains and heaters and pool sticks” as well as broken bottles (Ch. 2). They are violent affairs.
The problem is that the Socs will always have the upper hand, because they are the ones with the power. They have all the advantages. Even if they lose a fight or a rumble, it won’t matter. The police and society will see them as just letting off steam, because the greasers are losers and delinquents who will never amount to anything.
Johnny has a rough home life. His father beats him. He never fully recovers from getting jumped by the Socs.
I had seen Johnny after four Socs got hold of him, and it wasn't pretty. Johnny was scared of his own shadow after that. Johnny was sixteen then. (Ch. 1)
Not every Soc or greaser wants to fight. Ponyboy, for instance, does not want to continue the cycle of violence. Johnny is just defending himself when he carries a knife. They are in the park when a group of Socs attack them. Johnny kills one, Bob, in order to save Ponyboy. Neither Johnny nor Ponyboy wants to kill anyone. They just want to be left alone.
After Johnny kills Bob, things get worse between the Socs and the greasers. They blame Johnny for Bob's death. They do not realize that Bob was trying to kill Ponyboy, and Johnny was protecting him. None of that matters to them. All they care about is that Johnny killed on of their own. It does not matter to them that Johnny never wanted to fight anyway. Johnny and Ponyboy go into hiding in an old abandoned church. Johnny is severely injured helping save some children on a field trip from a fire. He later dies of his injuries.
Ponyboy is not the only one who doesn’t want to fight. Cherry tells Ponyboy about another Soc, Randy, who does not want to show up at the rumble.
"He's not scared. He's just sick of fighting. Bob ..." She swallowed, then went on quietly. "Bob was his best buddy. Since grade school." (Ch. 8)
Bob’s death meant something personal to Randy, but to the other Socs it is just another opportunity for revenge. They will continue the cycle of violence.
Johnny's death hit Ponyboy hard, and gives him a chance to re-evaluate his life. It also forces a fresh start. Pony is able to get out, using his intelligence to break the cycle for himself. It is a fresh start Johnny never got.
You could say the conflict is Johnny vs. society, with the Socs representing society. Society does not give Greasers a fair shake. Society judges them on superficial, unimportant things. The Socs do this with him and all Greasers. Johnny's fear of the Socs is also affecting his life, where he goes, his self worth etc.