Please note the term Cuticle is used interchangeably between human Anatomy to describe the area above the finger or toe nail, it is also used in Botany to describe the waxy covering of a leaf or thickened skin of fruit; additionally the term is used in Entomology to describe the covering of insects.
This answer relates to human Anatomy, please refer to the Related Question - What is a Plant cuticle for a Botanical definition.
The cuticle, or Eponychium (in a person) is the small area of skin above the finger- or toe-nail. It's what people sometimes bite, like the sides of their fingers. It is at the base where the nail connects to the actual finger. This skin is sensitive and delicate. It is a very thin layer of skin, and can be broken easily. Some people who bite their nails also tend to bite their cuticles. Despite being so small, they can bleed a LOT! It is important to keep up the condition of your cuticles. Many nails salons will do this (sometimes you might have to ask, though) when you get a mani or pedi done. This can be even done at home, because it just involves making sure the skin does not grow over the nail. That can cause some minor health problems, like irritation, slight pain, possible infection, or breaking of the skin. Cuticle health can either be maintained by trimming back the skin, or by pushing it back. This is most easily done when the skin is soft and moist, like after a hot shower or bath.