Answer:
To fix a leaking double handle faucet, you will need to remove the cartridge, compression/valve stem, or ceramic disc. This worn out component causes water to leak either from the faucet itself or under the handles. Remove this part from the faucet handle and match it up with a new one or replace the seals/o-rings/gaskets. Use a faucet repair kit when replacing seals or o-rings.
Explanation:
Turn both water supply lines to the faucet OFF.
Loosen the set screw on the faucet handle.
Take the faucet handle off.
Remove the retaining clip/nut.
Pull cartridge out. ( use pliers to pull up and out)
Drop in a brand new cartridge.
Install the retaining clip/nut.
Put faucet handle back on.