Full answer
Most faucet leaks come down to a single worn rubber component: an O-ring, washer, or ceramic cartridge that no longer seals tightly when the handle is closed. Identifying which type of faucet you have — ball, cartridge, ceramic disc, or compression — determines exactly which part to replace.
Before touching anything, shut off both hot and cold supply valves under the sink (turn clockwise until they stop), then open the faucet to release any residual pressure and confirm the water is off. This step is non-negotiable; skipping it means water everywhere when you remove the handle.
For a cartridge faucet (the most common modern type), remove the decorative cap on top of the handle, unscrew the handle screw, and pull the handle straight up. The cartridge is held by a retaining clip or nut — remove it, pull the old cartridge out noting its orientation, and push the new one in the same way. Reassemble, turn the water back on slowly, and test.
For older compression faucets, remove the handle and packing nut to access the stem. The rubber washer at the bottom of the stem is almost always the culprit — it costs about $1 at any hardware store. Having a good adjustable pliers or channel-lock set makes the job much easier and prevents scratching finish components.