Hairs are made of dead epithelial cells strengthened by a protein called keratin. Actually, epidermis (the upper layer of the skin) descend into the dermis (inner layer of skin) to form flask shaped structures. These structures hair follicles have a group of epithelial cells that form a bulb. Below this bulb is the dermal papilla or hair papilla which consists of connective tissue, supplied richly with blood vessels and connected by nerves. This region supplies nourishment to the root cells of the hair.
The epidermal cells of the hair root are very active and they divide rapidly. As new cells form, the old cells are pushed upwards in the form of hair.
Beyond the bulb region, the cells die and become horny with the addition of keratin. As the dead cells continue to add rapidly in the root, the shaft grows in length and pierces through the epidermal layer and projects out as hair. The hairs normally grow for about 18 cm in a year.
Thus the root is the only living part of the hair. Hence, when the hair above the skin is cut, we don’t feel any pain. But, if the hair is pulled out, because of the presence of the sensory nerves in the hair papilla, we feel the pain. A similar mechanism operates in the nail also. The root or matrix of the nail is made of soft epithelial cells and the growth takes place here as the cells are active and alive. As the cells grow the older cells are pushed out as the nail plate. Beyond the root region, the plate cells die and become horny again with keratin addition. Beyond the finger tip these horny cells project out as nails. From the root to the finger tip, the nail plate is connected to the underlying epithelial cells (nail bed), which are provided with blood vessels and nerves. When we pull this part out, we feel the pain, because the epithelial cells are also damaged. But beyond the finger tip, it is all dead cells and no pain is felt when cut. The nails grow to 
about 5 cm in a year.