This is what should be replaced. Do this by using a screwdriver to unscrew the end. Replace the washer, tighten the screw, the reinsert and screw back in both places where you loosened. Before tightening at the faucet base, use a caulking adhesive on the threads to better seal them. Turn the water back on and you should be in business.
More modern faucets require a different approach.
Use a utility knife to pop off the very top of the faucet. Once the handle is off, unscrew the nut underneath with your hand. This will allow you to remove the cartridge, revealing two washers. These two washers need to be popped out and replaced. Take these washers to the hardware store to find the proper replacements.