Final answer:
The fold of skin overlapping the side of the nail is called the nail fold, while the fold that meets the proximal end of the nail body is called the cuticle or eponychium.
Step-by-step explanation:
The fold of skin overlapping the side of the nail is called the nail fold. It helps to anchor the nail body in place. The nail fold that meets the proximal end of the nail body and overlaps the nail plate is called the cuticle, also known as the eponychium.