Cut around the damaged section of the carpet. Remove the damaged section by lifting at one end and firmly pulling upwards. Set the damaged section aside.
Place a piece of tape next to the hole in the carpet. Use it to show which way the nap, or the carpet fibers, lean. Draw an arrow on the tape to indicate this.
Place the damaged piece of carpet on a piece of new carpet, one which matches the old completely. Line the pieces up so the naps are flowing in the same direction. Cut around the piece of damaged carpet, through the new carpet, to create a replica.
Cut two strips of carpet tape to the length of the diameter of the cut hole. Remove the backing from both pieces.
Place the two pieces of tape in the carpet hole. Remove the top backing and discard it. Set the new replica of carpet in to the hole, ensuring the nap of the new piece matches the arrow of the general carpet.
Fluff the carpet, using your fingers. Pat down and run your hands along the carpet, in accordance with the nap's direction. Repeat all steps with each section that requires repair.