If you're tap has started to drip after use or you're finding it takes increasingly more work to turn it off, then chances are the tap's washer has worn out. The washer is a small coin shaped piece of rubber with a hole in the middle of it that compresses to form a tight seal when a tap is turned off. However over time, the rubber will begin to disintegrate and lose it's elasticity and this will cause your tap to drip.
In today's 'throwaway society', some people would sooner think about simply replacing a dripping tap altogether than replacing the small rubber washer, particularly seeing as the callout charge alone for a plumber will be upwards of £30 for such a simple task. But a new washer will cost you about 40p from your local hardware shop and will only take you about 15 minutes to fit.
Before you start any work with taps, be sure to isolate your water supply and drain the pipes. To do this you'll need to find your stopcock (usually under the kitchen sink in most old houses) and then turn on a downstairs tap to drain the water from the pipes. Also, remember to put the plug in the sink/bath you're working at to ensure you don't lose any small pieces (it happens).
To get into the tap, you're first going to have to remove the plastic cover at the top of the tap. This usually says whether the the tap is hot or cold either by colour or an H or C. You can do this by prying it off with a small flat-head screwdriver but be carful not to cause damage to the tap.
Under the plastic cap, you should find a screw which you'll need to loosen. This will enable you to remove the handle of the tap. Next, you're going to need to use an adjustable spanner to remove the main body of the tap (as shown in the diagram below). Be sure to hold the tap straight while doing this with a strong hand or another spanner to ensure that no damage is caused to the sink or the pipes beneath it by the forces required. It's also worth wrapping the tap with a piece of cloth while doing this part to ensure the adjustable spanners don't scratch the metal.
Now that you've taken your tap apart, you should be able to locate the black, rubber washer. Chances are it'll be secured be another screw to the main body of the tap that you'll have to also loosen. Replace the tap washer with your new one and then reassemble the tap making sure that everything is screwed securely but not excessively tight. Then test the tap a few times before turning the water supply back on.
Tap washers come in a variety of sizes so it's probably worth taking this washer along with you to your hardware shop to make sure you get the right type. Most hardware shops also sell boxes with a selection of washers in them for just a couple of pounds.
I stumbled upon this simple diagram online (which strangely enough was printed originally as a cigarette card) that shows which part of the tap needs to be removed, what the washer looks like (C), and how the washer actually works to turn the tap off. However, it is also a particularly old fashioned style of tap and has different components to more modern taps.