Laundry faucets usually don't have elaborate finishes with leak-proof ceramic disk valves. They are utilitarian more than stylish, and the ones with two handles almost always have compression valves, which are the most basic types of valves. When you turn the handle, a threaded stem tightens a washer onto the water inlet to stop the water. The most common cause of leaks is worn washers, and to replace one, you have to remove the stem.
The spout and handles of a double-handled laundry faucet are usually incorporated into a single body made of chrome or plastic. Pipes inside the faucet connect the handles to the spout, and these pipes rarely leak -- if they do, you need a new faucet. The valve inside each handle has an inlet hole that supplies water and an outlet that directs water to the spout, and the screwable stem controls the exchange of water between them. The washer on the end of the stem does all the work, and it's usually made of rubber and wears out easily.
Replacing the Washers
When the spout begins dripping, and you can't stop the drips no matter how tightly you turn the handles, it's time to replace the washers. The first thing you need to do is to turn off the water. The handles are usually held by screws that you can remove with a Phillips screwdriver. After unscrewing a metal collar to release the valve stem, you should be able to pull it out with pliers. The washer is on bottom of the stem, and it's held by a single Phillips screw. Quote the model number of the faucet to get the right replacement washer. It should be imprinted on the back of the faucet.
Stuck Handles and Stems
Repairing a faucet become more difficult when it's old and the handles, stems or screws are frozen. Using plenty of spray lubricant to loosen stuck screws prevents you from stripping them. You usually can't free stuck handles with lubricant, however, but you can tap them with a hammer or douse them with hot water. In a pinch, you can use a handle-puller, a corkscrew-like device available at hardware stores. When the stem is stuck, you can usually loosen it by spraying lubricant and wiggling it around with pliers. In some cases, you may need to rent a stem puller from a rental outlet.
Troubleshooting Other Problems
If you notice a loss of water pressure, try unscrewing the aerator from the spout. You'll probably find that it's filled with mineral deposits, especially if the faucet is old. You can scrape these off, or you can soak the aerator overnight in white vinegar to dissolve the deposits. If the faucet has a swivel spout, and water is leaking from the swivel joint, there's probably a worn O-ring. Remove the spout by unscrewing the collar that holds it with adjustable pliers, and when you get it off, the O-ring you need to replace should be apparent.