Most of the time, a leaking faucet is due to a damaged or worn out washer.
After time and continued use, the rubber washer can harden, tear, or lose its form which prevents it from adequately sealing off water flow.
Usually, the procedure is to tighten the faucet even more to keep it from leaking, but this only causes more damage to the washer and can also damage other internal parts.
It is recommended to replace the washer as soon as leaking begins under normal torque of the faucet handle.
To replace the washer, turn off the source of water to the faucet.
There is normally a shutoff valve located under kitchen and bathroom sinks, or, for a bathtub, it is located behind a panel, in a crawlspace, or in a basement (for first floor bathrooms).
You can also turn off the water supply at the main valve if it is easier to locate.
Find and remove the Phillips-head screw on the top of the faucet handle.
This is usually located under a cap in the center of the handle.
Once the screw is removed, pry the handle up and off the stem.
Next, unthread the stem by turning it according to the "on" direction.
If any metal chips are found or are created from working on the faucet parts, be sure to remove them as these can cause additional damage by the constant working of the faucet while in use.
The washer is located at the bottom of the stem and is held in place by a brass screw.
Remove the screw and the washer from the stem.
Take the washer with you to a plumbing hardware store.
Faucet washers come in a variety of styles and sizes so it's best to match it up correctly on the first trip.
Install the new washer and reassemble the stem.
Apply a small portion of petroleum jelly or silicone grease to the stem threads to make the faucet's action smoother.
Reattach the handle.
Turn the water back on at the source and open and close your faucet to test it.
Problem solved.