If your double-handled bathroom faucet is dripping, grab an adjustable wrench, a Phillips screwdriver and a flat-head screw driver. Then head down under - under your sink, that is.
Step 1: If life is good, there will be two shutoff valves, one for the cold water and one for the hot water. Turn the water on in your sink, reach under and twist each valve handle counter clockwise until the water flow has completely stopped.
Step 2: 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. Use an adjustable wrench to remove the retaining nut or bonnet that holds the faucet stem in place. Place the handle back on the stem and turn the stem in the same direction you would to shut off the water. With a little luck the stem should come out easily.
Step 3: If your stem appears to have the remains of a flat rubber washer on the end, held in place by a screw, you will need to replace the worn washer with a new beveled head washer. Otherwise, you will need a new set of seats and springs, which are located in the faucet base just below where the stem sat. If this is the case, 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 your screwdriver will usually do the trick in a pinch. The spring is located beneath the seat.
Step 4: Take your stems, seats and springs to the hardware store and find the necessary replacement parts. If your stem is worn or cracked you should replace it while you have the chance, otherwise a set of 30-cent washers or a pair of $3 seats and springs will solve all your problems or at least the immediate one at hand.
Step 5: Once back home, install your new seats and springs. If appropriate, replace the stem, hand-tighten the retaining nut. Do not use the wrench to tighten the nut or you may break something. Screw the handles back on, pop the buttons in place, open the shut off valves and pour yourself a nice tall glass of water: or something stronger if you so desire.