Many modern taps are fitted with ceramic disc cartridges rather than traditional washers on spindles and they work in a different way. These ceramic discs are often referred to as being maintenance free but of course they can become damaged and scratched which will mean they need to be replaced.
Ceramic disc cartridges are made with two ceramic discs each with two holes in it. One disc moves with the tap handle and the other disc remains in position. When these holes are aligned, the water is allowed to flow. In the off position, the holes are no longer aligned and the water stops.
Ceramic cartridge type taps can leak or drip and this is an indication that the discs have become damaged. Unfortunately, these cannot be replaced so a new cartridge will need to be fitted.
It's important to note with these that they are left and right handed so you will need to be sure to buy the appropriate one. Take the old one with you to make sure you buy the right one to fit your particular tap.
Cleaning and replacing the cartridge
Turn off the hot and cold supplies and open the tap to drain off the residual water. Place the basin / bath / sink plug in the plug hole. You'd be surprised how small components are attracted to waste outlets !
Remove the tap heads. These are sometimes push fit or may require the removal of a retaining screw. Using a spanner, undo the headgear of the tap. Hold the tap firmly at the same time to prevent this being rotated by the force. Remove the cartridge. As mentioned before, they are 'handed' so, either work on the taps one at a time or, make sure you keep the cartridges separate.
Inspect the cartridge for damage and debris. If they are simply dirty and not worn, then a clean with a damp rag should suffice. If they are damaged, cracked or worn they will need to be replaced.
You should also check the rubber seal for damage. These can cause dripping from the base so it's worth replacing them while you have the assembly apart.
With either the cleaned or the new cartridge, reassemble the tap. Fit the cartridge into the tap assembly and refit this into the tap body, tightening gently with a spanner.
On completion, switch the hot and cold water supplies back on and check the function of the new or cleaned ceramic disc cartridges.