The conflict between the greasers and the Socs is that they are rival gangs that are constantly fighting each other.
As with most rivalries, there is always an old grievance to nurse. The greasers are the working class kids and the Socs are the wealthy kids. They are always fighting each other in group brawls called rumbles. They are not fighting for any particular reason other than the fact that they have been fighting for so long that they have to get back at each other for the last fight. So there is always a new reason to fight.
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)
Pony explains that the greasers, who wear their hair long greasy and usually dress in t-shirts and jeans, often get into trouble by robbing gas stations or wrecking houses. They are not usually real criminals though. They are just poor kids who are sticking up for each other.
The Socs, on the other hand, are the rich kids. Pony calls them the “jet set.” They are the socialites from the social club. They are just making trouble for fun or to stave off boredom until their trust funds kick in and they go off to college.
The greasers and Socs are constantly fighting each other. Socs will jump greasers in groups, just because they are there. Then a group of greasers will have to get together and fight the Socs in a rumble, to make up for it. It continues like that. Pony explains his perception of the Socs’ behavior.
[The] Socs had so much spare time and money that they jumped us and each other for kicks, had beer blasts and river-bottom parties because they didn't know what else to do. (Ch. 3)
Though Pony is a little more empathetic than most greasers, and he does begin to accept that the Socs have feelings too, and do have some problems. Some of them do not want to fight. Randy tells him later that he has seen too much violence, especially after Bob dies, and will not participate in the rumble. While the Socs can walk away, most greasers cannot. Pony is one of the few who is intelligent enough and sensitive enough to have the option. He finally does take it, after he has seen too much death himself.