A dripping faucet wastes water and the steady drip can stain fixtures. The repair depends on the faucet type, but all start the same way: shut off the water and plug the drain so you don't lose small parts.
Shut off the water
Close the shutoff valves under the sink, then open the faucet to release pressure. If there are no local valves, shut off the main. Plug the drain with a rag.
Identify your faucet
Compression faucets (separate hot and cold handles) usually leak because of a worn rubber washer or seat. Cartridge, ball, and ceramic-disc faucets (single handle) usually need a new cartridge or O-rings. Take the old part to the store to match it.
Replace the worn part
Disassemble the handle, replace the washer, O-ring, or cartridge, and reassemble. Smearing plumber's grease on O-rings helps them seat and last. Turn the water back on slowly and check for leaks.