When a double handled bathroom faucet is dripping, grab an adjustable wrench, a phillips screw driver, a flat head screw driver, and head down under; down under the sink that is.
Step 1: If life is good, there will be two shut off valves, one for the cold water and one for the hot water. Turn the water on in the sink, reach under and twist each valve handle to the left until the water flow has completely stopped.
Step 2: Next, remove the faucet handles by prying up the plastic or metal button on the top of each handle and removing the handle screw. Pull the handle straight up and off to expose either a metal or a plastic faucet stem.
Step 3: Use an adjustable wrench to remove the retaining nut or bonnet that holds the faucet stem in place.
Step 4: Place the handle back on the stem and turn the stem in the same direction that would to shut off the water. With a little luck the stem will come out easily.
Step 5: If the stem appears to have the remains of a flat rubber washer on the end, held in place by a screw, replace the worn washer with a new beveled head washer.
Step 6: Otherwise, replace the seats and springs, which are located in the faucet base just below where the stem sat. Use a towel or wash cloth to soak up the excess water and pry the little black rubber circle up and out of the base. A dental pick works best, but a screw driver will usually do the trick in a pinch. The spring is located beneath the seat.
Step 7: Next take the stems or seats and springs to the hardware store and find the necessary replacement parts. If the stem is worn or cracked it should be replaced, otherwise a set of beveled washers or a pair of seats and springs will solve the problem.
Step 8: Once back home, install the new sets and springs or the new beveled washer, replace the stem, and hand tighten the retaining nut. Do not use the wrench to tighten the nut or something may break. Screw the handles back on, pop the buttons in place, open the shut off valves and pour a nice tall glass of water… or something stronger if desired.