Remove Old Washer
Find the shut off valve under the sink, and turn the valve handle clockwise to shut off the water supply to the faucet. Turn on the faucet handles to release any residual water from the line. Turn off the handles.
Remove the cover on the top of the handle with a utility knife. Insert the blade of the knife between the edge of the cover and the handle, and pry the cover off. Remove the retaining screw in the center of the handle with a Phillips head screwdriver. Pull the handle off the faucet.
Grasp the nut securing the faucet stem to the faucet body with a pair of slip joint pliers. Turn the nut counterclockwise to remove the nut and the stem. Unscrew the stem from the faucet nut with your fingers.
Turn the screw securing the rubber flat washer to the bottom of the stem with the Phillips head screwdriver. Pull the washer off the stem and remove the O-ring from the outside of the stem. If necessary, cut the O-ring with the utility knife to remove it from the stem.
Take the washer and O-ring to the hardware store for replacements. Ensure that you obtain replacements that are the same diameter as the existing parts.
Install New Washer
Roll the O-ring over the outside of the stem until it rests inside the groove. Inspect the O-ring to ensure that there are no twists.
Position the washer on the bottom of the stem, and secure the washer to the stem with the retaining screw. Thread the faucet nut onto the stem until it is hand tight.
Insert the stem into the faucet body, and thread the faucet nut into the faucet body. Tighten the faucet nut another full turn with the slip joint pliers.
Push the faucet handle over the stem and secure it with the retaining screw. Place the cover on the top of the handle and push down until it snaps into place. Turn on the water supply to the faucet.