This step by step guide relates to a conventional pillar cold water tap, rather than a mixer or wall tap. The tools you will need are a screwdriver, an adjustable spanner, multi-grip pliers, a new washer or a washer and jumper unit.
Taps usually leak for three reasons: a faulty washer, a worn seat, gland packing/O rings need replacing. Make sure you buy the right washer and o-ring for your tap. The first thing to do is turn off the water supply, either at the isolating valves under the taps, but more likely at the main stopcock. Open the tap full on to drain away any water left in the pipe. It's a good tip to put the plug in the sink so that you don't lose any small parts down the plughole.
If the tap has a cover, remove the coloured red or Blue /hot or cold indicator button, undo the head retaining screw beneath and lift off the handle. If the tap in question doesn't have a cover, the nuts are more easily accessible. The main valve nut should be undone with an adjustable spanner. Another tip, to protect the nut against scratches when removing it, is to place a cloth between the nut and spanner. It's very important to hold on to the tap while undoing the nut, to stop the body of the tap turning rather than the nut. If this happens it can lead to leaks in the pipe work. Lift out the whole upper part of the tap, and remove the spindle or jumper value that carries the washer. Check the washer next. If it looks old and rough there is a good chance this will be the cause of the leak. If this is the case remove the old washer. This may be held in place by a nut, so hold the jumper valve with a small pair of pliers and undo the nut with a small spanner, and put on a new washer. You can use a new jumper valve with a washer already attached. Check the washer seat for wear, which will stop it making a perfect seal. A plastic seating grommet may be pushed in to make a new seat. Push the grommet into the hole. Replace the tap assembly in reverse order and screw the handle back down. This will bed in the grommet and a new seat. If the grommet is too large or small, the tap may leak.
If the tap is leaking from the spindle, the O ring could need replacing. Remove the spring or circlip from the spindle, screw out or tap out the spindle and replace the o ring. If you have any doubts at any stage of your DIYrepairs, consult a qualified plumber.