The washers in a tap are subject to constant pressure and friction as well as hot water, in the case of the hot tap. This breaks down the rubber or plastic that the tap is made of and will result in dripping. Not matter how hard you turn that tap handle, you can't quite get it to shut off. The washers need to be replaced. This is a quick five- or 10-minute job that will keep the dripping from annoying you and wasting water. When you pick up washers at the store, get extras, then it is always an easy fix if another tap develops a leak.
Turn off the water at the main shut-off valve or under the sink if there is a shut-off located there. Turn on the tap you are working on to release any water stored in its line. Place the sink plug in the drain to prevent the screw and other items from dropping down the hole.
Use a flathead screwdriver to pop off the clear plastic top on the tap. On some tap units the washer sits on top; others have a housing you need to remove. It should pop up and underneath is the screw for the tap. Unscrew it and pull up on the tap to remove it.
Hold the base of the tap with one hand and use a spanner, crescent or adjustable wrench to undo the brass or stainless coupling at the top. Pull the valve out of the tap and pull off the old washer on the bottom. Replace it with a new one. Put the valve back into the tap and tighten the coupling with a wrench.
Put the tap housing back on and tighten the screw. Replace the insert if there was one and then the cover on the tap. Turn the water back on and then turn on the tap. Check for leaks. If there is a problem, take apart the tap again and try with a different washer -- there may have been a defect in the one you used. Make sure all the parts are tight because loose components could cause leaking and keep the tap from turning off properly.