Most older two-handle faucets have stems that move up and down as the handle is turned. A rubber washer at the bottom of the stem presses against a seat in the faucet body to seal out water. If the washer or the seat becomes worn, water seeps through and drips out of the spout.
If water seeps below the handle or the base of the faucet, an O-ring or the packing probably needs to be replaced.
Getting the parts
All-purpose repair kits for stem faucets contain washers and O-rings of various sizes. To make sure you have a perfect match, take the stem along when buying the repair kit. An older type of faucet may need a packing washer or string packing. If the stem itself is worn, replacing the rubber parts will not solve the problem. You can replace the stem if you can find a new one; replacing the faucet may be the best option.
Less than an hour for most repairs
Screwdrivers; adjustable wrench; perhaps a handle puller, seat wrench, or seat grinder
Shutting off water, dismantling a faucet, installing small parts
Shut off the water, close the sink stopper, and place a rag in the sink to catch any parts
Washer, O-rings, silicone grease, packing