Turn off the water supply
Turn the water supply off at the meter (usually located near your front fence). If you need to drain a hot water cylinder to work on a hot tap, make sure the heating is turned off first.
Drain the pipes
Turn the leaking tap on to drain the pipes.
Remove the tap cover
A traditional style tap will have a cover that needs to be removed. The cover may actually be the handle or the handle may be separate. There is usually a small screw on the top of the handle, usually under the plastic indicator buttons which lift off. (Alternatively it may be a steel cap that unscrews).
Cross top taps will also have a shield beneath the handle that needs to be removed. It should unscrew anticlockwise to reveal the inside of the tap. You should be able to unscrew this by hand, but if you need to use pliers wrap a thick cloth around it first to prevent damage to the tap.
Unscrew tap spindle
Unscrew the tap spindle from the body with an adjustable wrench. This will reveal the washer which is held on by a small nut. Undo this and attach a new washer. Then replace the whole top assembly in the reverse order that you removed it. Turn the tap off and turn the water back on at the meter.