A dripping faucet can be annoying, and it wastes water. Fortunately, it can be fixed by pretty much anyone with a few tools and about an hour to spare. Of the four types of faucets you might find in your bathroom, only one has washers that might be worn out, and it always has two handles. If your faucet has a single handle, you are probably dealing with a cartridge or ball-valve faucet since the third kind, the disk faucet, rarely leaks. You can service all three types of faucets quite easily.
Turn off the water to the faucet by turning off the angle stops underneath the cabinet.
Unscrew the nut holding the handle to the faucet stem and remove the handle. You will need either a screwdriver or an Allen wrench to do this, and the nut is either on the top of the handle or hidden somewhere underneath.
Remove the cartridge, if it is a cartridge-type faucet, by unscrewing the holding nut with channel-locking pliers or removing the holding pin with needle-nose pliers. Pull the cartridge straight out with pliers, or screw the handle back on and use it to pull the cartridge out. Note the orientation of the cartridge in the faucet body because it will have to be replaced in the same way.
Examine the O-rings on the cartridge and replace any that appear worn. If they do not appear to be damaged, check the valve seat for signs of chipping or corrosion. If the valve seat appears intact, buy a new cartridge. Reassemble the faucet by inserting the cartridge in the same orientation as it was when you took it out. Then replace the holding pin or screw on the holding nut and replace the handle.
Follow a similar procedure for a ball-valve assembly. Unscrew the holding nut with channel-locking pliers and remove it. Remove the ball valve and the springs and examine the valve seat for signs of wear or corrosion. Check the seals around the water inlets and replace them if they are worn. Replace the ball valve if it is chipped or badly corroded. Reassemble the faucet by setting the ball valve and spring in place, being sure the valve is seated correctly in the water inlets, and screw on the holding nut. Then replace the handle.
Repair a disk faucet by removing the handle and unscrewing the disk assembly, then lifting it out. Examine the three O-ring seals in the inlet valves and replace them if worn. If that doesn't stop the leaking, replace the entire disk assembly.