Salon Life's cuticle explanation video
This is the best image and video I could find online that is accurate, and I don't want to publish them here and take traffic from them, so please visit the links. :) Thanks!
See the Proximal Nail Fold (PNF)? That is what most people confuse with the cuticle and they end up doing damage to it. PNF should never be cut, poked or prodded aggressively on purpose. It works as a seal to protect you from bacteria and infection and needs to be allowed to do its thang. :)
The cuticle is an epidermal layer of skin that is virtually invisible. It grows out in a thin layer attached to the top of the natural nail plate from the underside of the PNF.
What most people call the cuticle is actually the PNF. Cutting, picking, or any other attempt to remove the PNF should be avoided at all costs. Doing so greatly increases the risk of paronychia (a nasty painful bacterial infection). On top of the dreaded infection, the body replaces the cut/torn/eradicated skin with scar tissue and/or calloused and hardened PNF so it will be even worse (bigger/tougher) when it regrows.
If you or someone you love cuts their PNF, get them to stop. Have them use softening and exfoliating products like Alpha Hydroxy creams, CND Cuticle Eraser, and pure lanolin to soften and exfoliate the hardened skin. They can even use a gel or creamy cuticle remover, which will help to remove the dead cuticle from the nail plate and detach the PNF from it with very gentle pushing. My favorite remover is Blue Cross from Sally Beauty or KBShimmer Clean Start gel cuticle remover, and I used to use Sally Hansen's Instant Cuticle Remover in the blue bottle which is also really good.
I took the below photo of my own nail to demonstrate the part of the cuticle that can safely be removed weekly or bi-weekly to keep your nails pretty and your polish on. :)
The purple arrows shows the stuff you will want to remove, the red arrow is what you want to leave be (PNF) except to detach it from the dead cuticle so that it will retreat back where it should be. When PNF gets pulled down the plate by the growing cuticle, it will stretch the PNF out and make it look bigger than it is. Detaching it will let it 'spring' back and after a day it will look like it should. To see how this is done please visit The Salon Life on youtube, she has dozens of videos of natural nail manicures that teach how to do this gently.
Usually, the purple arrow portion is not visible but thanks to acetone it showed so I could photograph it for you!
The red arrow is the PNF which seals out bacteria to prevent infection, and is the part you'll want to moisturize daily and let it do its thang.

Go forth and treat your PNF well!