A dripping faucet is most likely the fault of a worn-out washer. To replace it, first turn the water off under the sink. Then, drain the faucet and carefully take it apart, keeping track of the order and orientation of the parts. Remove the worn-out washer at the bottom of the stem, and bring it to the store to get the correct replacement size. Put it all back together, and the grating drip should stop. For a washerless faucet, check out this how-to to diagnose and fix the problem.