The hair and nails are composed of dead cells. These are made up of a dead protein called 'Keratin'. Because they are dead tissues they do not have supply of nerves. Nerves are the organs which carry (convey) sensation of pain from that part to the brain. So the sensation of pain does not reach our brain and we do not feel pain when hair or nail is being cut.
But when we pluck the nail or hair we do feel pain because the nail bed and hair root have well supply of nerves