What Is This Thing Under My Toenails?

I have the type of toenails where you can see really well under it; unlike where many people have toenails that are almost totally closed so that you cannot see underneath the toenails. Under each of my toenails, there is a whitish ball of dry skin stuck to the bottom. It is very sensitive, so it kinda hurts when I touch it. It also prevents me from cutting my toenails correctly; it REALLY hurts when the clipper touches it. So what is this white ball of dry skin? How do I get rid of it?