As a nail enthusiast, I feel I must respond to this...
IMPORTANT STUFF. Removing old, dead cuticle skin is part of good nail care. Think of it as exfoliating! You are helping to allow the new healthy cuticle to breathe and to do it's job.
And that is the important part:
do it's job! Your cuticle is the "gasket" between your flesh and your nail. Your cuticle prevents moisture, bacteria, etc from getting into your body at the nail. It's like the caulking around your windows keeping the rain outside.
Never, ever remove living healthy cuticles! You are cutting into living flesh - and that should only be done to your body by a doctor, using STERILE instruments, not just "sanitized" instruments (salon's instruments are merely sanitized).
If you have any doubt, use a cuticle stone soaked in water to remove dead cuticle skin. (use Google Image to search for "cuticle stone" and you'll get lots of pics) The stone gentle scrubs old skin off, but leaves living skin untouched.
Sorry for all the bold print, but bacterial infection (what we nail enthusiasts call the "green meanies") is NOT GOOD.
AT