To repair a leaky faucet, remove the faucet's handle knobs, unscrew the packing nut, and take the stem out of the faucet. Locate the O-ring and the washer in the valve seat, and examine them. These are the parts that usually cause a faucet to leak. Install a new washer and O-ring, and reassemble the faucet in the opposite order.
Make sure that the water is shut off before taking the faucet apart. This includes shutting the water off at the handles underneath the sink. To take handle knobs off the faucet, use a flat-head screwdriver to pry them up, and unscrew the screws that secure the handle to the faucet. If the screw is stuck, apply some penetrating oil to it to loosen the screw.
Be careful when taking the stem out of the faucet, as some of them need to twist to remove, while others are easy to remove without it. Make sure that none of the parts become damaged in the removal process. Washers and O-rings come in different sizes, so make sure the parts fit the faucet correctly. Otherwise, the faucet may start leaking again or simply stop working. It may be a good idea to take the old O-ring and washer out of the faucet, and take them to the hardware store when buying new parts.