You don't say anything about what kind of faucet you have, so this may not apply. But if yours is the kind with handles you turn to control the water, you probably have worn-out compression washers. In this kind of faucet, water is shut off by pressing a rubber washer on the end of a threaded stem against an opening in the faucet body. When you turn the handle, you move the washer away from its seat and the water can flow. Over time the washer gets worn and no longer seals off the water. Also, the valve seat the washer presses against can wear and allow water to sneak past the seal. The result is that annoying drip or a steady trickle.
This site, among many others, shows how to change a washer and fix most drippy faucets. If you search for "repair a leaky faucet" you'll find lots of sites to pick from, including many with videos. If you faucet is not a compression type, a single-lever cartridge for instance, there are links to other repair procedures.
If your faucet is old and the parts very badly corroded and worn, you might want to fix the leak by replacing the entire stem as a unit. You can find replacement stems for most brands of faucet at big-box home improvement stores and at many hardware store. It's a little more expensive, but often much easier to unscrew the old one and pop in the new than to start trying to get the old washer off the stem.
A last tip: Whenever you do any work on a faucet, remove the aerator from the spout and run the water to help clear the line. If you just turn the water back on and run the faucet, debris that was kicked up during the repair can clog the aerator screen. Then you're left wondering why you get only a weak trickle or no water at all after you fixed the trouble.
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