This should be done promptly to avoid unsightly stains on sinks/baths because the problem will get worse and the seating of the taps will require replacement.
The majority of dripping or leaking taps can be solved by simply replacing the washer - it is a small rubber ring that releases water only when the tap is open, thus preventing leaks. Firstly, close the main valve - the last thing you need is a waterlogged bathroom. Next, turn the tap knob anti-clockwise, till it loosens. You might have to yank it off towards the end.
Now use a spanner to unscrew the nut that holds the washer in place. Remove the washer and take it to the hardware store as a reference.
Get a replacement and fix it on - layer by layer - just as you removed it. Make sure that pieces of the old washer are not left in the seat of the tap before finally reassembling it and switching the water back on. Once again, remember to close the main valve into your bathroom or kitchen before you do any work.
If the leak is occurring from around the spindle of the tap or the tap is juddering during use then this can often be resolved by replacing the gland packing or O-ring seals in the tap.
If a tap continues to drip once the washer has been replaced then it will probably require reseating. Reseating is required because grit or general wear and tear, has caused the seating to become worn, this can be remedied by reseating the tap. .
Some older taps which are brass, may suffer from old age by developing cavities and becoming porous. If this is the case then the taps will require changing.
When repairing a faucet, line the sink with a towel to keep tools from damaging the basin. This will also keep small parts from sliding down the drain.