It sounds as though a new washer is required.
To turn off the water, you may have an isolator valve. If you look at the copper pipes under the sink that go up to the tap, you may find a fatter section fitted in, on both the hot and cold pipes. There will be a flat head screw in the middle of each section. Turn this clockwise to stop the flow of water. This saves using the main stopcock. Here is a picture of an Isolator valve with an arrow pointing to the screw.
http://www.plumbersinbirmingham.net/images/iso lation_valve.jpg
You may as well replace both washers whilst you are there.
Remove the disc in the centre of the taps (the bits you actually turn to turn the water on/off. The discs are pushed in and a small flat blade screwdriver should prise them out.) In the centre should be a srew. Undo that and the tap head should come off. Underneath you will see a stubby splined rod and a big nut that screws it into the tap. Undo the nut. It may be tight so hold the whole tap assembly to stop that moving.
Once this is out, you will probably sww a chewed up rubber washer. There is a hole in the middle of the washer. Remove the old washer and push on a new one of the correct size. (You may need to take the old one to B&Q or wherever to compare sizes). Check no rubber is left down in the cavity in the body of the main tap assebmbly.
Reassemble. Do not ridiculously overtighten the nut as you put it back together as you may strip the thred. Otherwise put it all back together in reverse order and turn the water back on at the isolator valves.
If the tap still drips after all of this, the seating may need re-grinding. If that;s the case come back and ask for more help.
This may help. The process is similar.
10:58 Tue 02nd Dec 2008