Repairing a dripping compression faucet involves replacing the washer on the end of the faucet stem and possibly cleaning the faucet seat. The faucet stem normally comes right out once you have the faucet handle and packing nut off the faucet. If your faucet is an older one, you may find that the stem is stuck and difficult to remove. A stuck stem requires a little bit of cleaning and some help from the faucet handle.
Mix a 50/50 solution of white vinegar and warm water in a small container. Brush the vinegar mixture into the faucet cavity with a small toothbrush. Allow the solution to work for 30 minutes. The vinegar penetrates the sediment that is "freezing" the stem to the faucet seat.
Place the faucet handle over the stem. Do not put the securing nut in the handle.
Turn the handle counterclockwise until it locks onto the stem and begins turning the stem. Once the stem is loose, pull the handle off the stem.
Finish turning the stem counterclockwise until you remove it from the faucet.
Inspect the faucet seat and repair as necessary with a seat tool. For stuck shower faucet stems, use a deep well socket or faucet stem removal tool available at home improvement centres.