To fix a dripping faucet, dissemble the faucet, replace the O-ring and washer, and reassemble the faucet. If this doesn't work, you may have to tighten faucet parts, replace worn-out seals, clean a corroded valve seat, or fix the plumbing.
Before fixing a dripping faucet, turn off the water supply from the main water line and the faucet itself. Gather an adjustable C-wrench, a Phillips or flat-headed screwdriver, and penetrating oil. Use the flat-headed screwdriver to pry off any decorative handle pieces, and use the same screwdriver to loosen the screws that secure the handle to the stem. Remove the handle, using penetrating oil if necessary to loosen the pieces. Loosen the packing nut with a wrench, and remove the stem, checking all parts for corrosion or damage and replacing them if needed.
If all pieces are in good condition, replace the washer in the valve seat. Purchase a new O-ring if the old one doesn't fit the new washer. Reassemble all parts in the proper order, tightening them securely. Turn the water supply back on to test the faucet. If it still drips, contact a professional plumber to diagnose and fix the issue. If left unfixed, a dripping faucet can lead to more serious plumbing issues down the road.