Ponyboy’s hair symbolizes his status as a greaser and is part of his sense of identity, which is why the Socs target him and why their threat to cut it is such a big deal.
Ponyboy’s hair is a big part of his identity. One of the first things Ponyboy mentions about himself is his hair.
My hair is longer than a lot of boys wear theirs, squared off in back and long at the front and sides, but I am a greaser and most of my neighborhood rarely bothers to get a haircut. (Ch. 1)
Ponyboy says he looks better with long hair, but it is more than that. The long hair is a class symbol. It associates him with the greasers, and adds to his sense of belonging. In his neighborhood, it is important to belong to one of the gangs. Also, having no parents and limited family other than his brothers, the greasers are part of his identity.
When the Socs target him and threaten to cut his hair, they are targeting his class. It is an attack not just on Ponyboy, but on the greasers. The threat against his hair is a threat against the greasers, part of the larger class war. Not being able to afford haircuts might have originated out of necessity with the greasers, but it became part of their identity along with blue jeans, t-shirts, and leather jackets.
Pony is afraid that far from just cutting his hair, the Socs really will begin the haircut “just below the chin” and kill him (Ch. 1). He knows that the battle between the classes is not all fistfights. The Socs are dangerous, and Ponyboy only stands a chance with the rest of his gang there to protect him. On his own, he is in great danger. Fortunately, at this point the rest of the greasers arrive in order to help Ponyboy and scare the Socs off. This is only the beginning of Ponyboy’s trouble though. Whenever they are not with him, he is vulernable.
In Ponyboy's world, his identity is the most important thing in his life. It's what defines him as a person, it's what keeps him safe, and it's what binds him with others in his gang. The greasers are largely defined by their choice of clothing and hairstyles, and without that definition, Ponyboy wouldn't be who he is, at least in his mind.
His hair symbolizes the few choices he can make in his own life that help define who he is as a person. He doesn't have money or other status in society, but he does have a gang that gives him an identity and self-worth. His hair is a symbol of both his independence (he chooses to be in the gang and not to cut his hair), and his belonging.
The Socs threat to cut his hair is a direct challenge to Ponyboy's basic identity, and thus is a very serious threat indeed. They are saying to him that without his "look", he is nothing, and Ponyboy is afraid of losing his identity.