Repairing a dripping tap is generally a very simple job and one that anyone can do as long as they apply a little common sense. This guide covers replacing the washer in a fairly standard tap design, the pillar tap, and gives a little other advice as well. Mixer taps, where there are two taps, are a little different but the general principal is the same except you normally have to remove the plastic handle (this bit you turn) first to get at the valve. To remove the handle you normally have to lever off a cover to expose a screw which holds it in place.
The first thing to do is turn off the water supply. If you are really lucky you will have isolating valves under the taps which will isolate the tap or room. If you are unlucky you will have to turn off the water at the main stopcock. In this guide I am fixing the hot taps so I only need to turn off the hot water. Locate the hot water stopcock (probably in the airing cupboard) and turn clockwise until it stops. Don't force it or you may break the valve.
If you live in a hard water area and you don't work your stopcocks two or three times a year chances are it is jammed. Ours hadn't been turned for 10 years I would guess and jammed about 90% of the way closed. If this happens don't force the tap. Apply a good squirt of WD40 or other penetrating oil and wait for 10 to 15 minutes and try again. If it moves and then sticks again repeat the oiling. Some taps, unfortunately, just get jammed open and there is not much you can do about it and it is a much more difficult job to replace them.
The tap I am repairing here is a particularly simple pillar style tap as it doesn't have a cover like most so the nuts are easily accessible. There are two nuts that can be undone. The one we are currently interested in is that main valve nut marked A the nut marked B is the gland nut and at present we aren't interested in this one (I have actually undone both in the photo but you only need undo A).
If you have tap like the one shown with the main valve nut on display you probably want to protect it when you remove it. The best way to do this is to slide a cloth in between the nut and the spanner as shown in the photo below. It makes undoing the nut (and tightening it again) a bit more awkward but it does prevent scratches.
Notice that I am holding onto the spout of the tap while I am undoing it. It is important that you do this or there is a danger you will turn the body of the tap rather than the nut. This can lead to all sorts of problems not least leaks in the pipe work under the skin / bath and a cracked basin.
Now there the valve is out check the washer. If it looks old and squashed and a bit rough then there is every likelihood that it is the cause of the leak. If it looks in tip top condition then the leak may be caused by damage to the valve seat which is the bit in the tap body that the washer presses on to. The valve seat is shown in the out of focus photo below. This is looking straight down into the tap. The seat should be shiny, smooth and free from scratches. Scratches are caused by little bits of lime scale that get trapped between the washer and the seat and grind away at it. If the seat is damaged it needs to be reground which is not a difficult job.
Since washers are very cheap it is worth replacing it even if it looks fine. The picture below shows a newly replaced washer. Washers are held on in one of two ways. Either by a small nut (which is invariably stuck due to lime scale) or a fixed pin. I'm not sure which is worse. If you can get the nut free replacing the washer is trivial if the nut is stuck though it's really hard. The pin system on the other hand might have less to go wrong but replacing the washer can be a challenge. The problem is that in order to hold the washer tightly the hole in the centre of the washer appears to be to small to go over the pin. The best advice I can offer is get someone strong in to help you and put some soap on the pin. I ended up using two grips to force the washer over the pin.
Now that the washer is replaced its time to reassemble the parts. Grab your spanner and set to work. Don't forget to hold onto the spout of the tap while tightening the valve. Once the valve is tight and the tap is in place turn on the water supply again. I suggest that at first you turn it on just a quarter or half turn and then go and check for leaks. If you don't see any turn it on full and then back half a turn. The half turn closed helps stop the stopcock from freezing (sticking).