As Pete said it's usually an Allen key.
Normally loosening this will allow you to remove the handle leaving the valve in place, if it's a newish tap then that valve is likely to be ceramic rather than a replaceable rubber tap washer.
The leak may be just that the valve is slightly loose in which case tighten with an adjustable spanner or the rubber O ring is broken or worn in which case you need to replace although a temporary repair can sometimes be made with PTFE tape.
If the tap is also dripping from the spout then the ceramic valve / rubber tap washer needs to be replaced.
Can get the valves at plumbers merchant or B&Q but best to leave the water off and take the old valve with you.
If just removing the handle I personally wouldn't bother to turn off the water but in your case It might be advisable.
EDIT:
Before turning off the stop **** look under the tap to see if there are isolating valves and turn off both hot and cold if it's a mixer tap. That means only the supply to that tap is turned off leaving the rest of the house unaffected.