The inner personal feelings of a 'cutter'
Cutting is an interesting thing. It seems so self-destructive, so hateful to oneself. Why would anyone want to scar their bodies, make taut, shiny white marks across perfectly formed skin? Why would anyone want to purposely cause their own blood to drip and puddle on the floor, drop by agonizing drop?
Because it is a form of control.
Everything else is spinning out of control: you can't control your parent's fighting, the stress of schoolwork, your job, maybe the way the kids at school treat you -- just life in general. You can't control the way you react to situations, the strength of your anger or joy or inner pain. You can't control your fears and ecstasies. You can't control any of this, but there is one thing you can control: the razor.
You can control how deep it slices, and how much physical pain you can handle. You can control how long you have to deal with it, or how often you do it. You can control the healing process: bandaging it up and hiding it, or letting the wounds show and watching the reactions of people around you. This is yours, and yours to control.
That is why people cut.
It is impossible to force a person to stop doing anything, especially anything that they may find empowering (no matter how destructive it actually may be). Therefore, the only way a cutter will stop cutting is if they come to the conclusion that it is harmful to themselves and choose to do so of their own will.
This does not mean nagging or guilt-tripping a cutter. This does not mean crying or sobbing about how their actions are hurting you. This does mean that (if they are underage) arranging for a licensed, well-regarded therapist and perhaps accompanying psychiatrist may be helpful.
Do not be a nosy parent and try to find out what goes on in the sessions; if the therapist is at all worthwhile, they won't tell you. There are confidentiality laws, and a patient-doctor trust that should not be violated.
If they are an adult, this means loving and supporting them regardless of the situation, and doing what is best for YOU -- just as you would in any relationship that you found had it's negative points. Weigh what you find to be most important, and how this measures on that scale. Is it worth the fight?
Cutting can be a form of self-identification. It can be a way of controlling your emotions; a way of controlling a world that seems uncontrollable. It's something that's not so terrifying in a world that is very terrifying.