(Using Perrin and Rowe Mayfair for reference)
Step 1: Turn off your water supply, using the isolation valves under the sink
Step 2: unscrew the shroud using a wrench, be careful not to damage/scratch the shroud (if the tap doesn't have a shroud, this wont apply, unscrew the handle)
Step 3: Take the shroud/handle off of the basin
Step 4: Unscrew the tap cartridge from the basin.
Step 5: Remove the tap cartridge. The cartridge may have a plastic external head, if so remove it and place it onto the new cartridge.
Step 6: Screw the new tap cartridge into the tap.
Step 7: Place the shroud back on top of the new disc.
Step 8: Screw the shroud into the disc.
All you need to do now is turn the water back on, give it a go and voila! quick, simple, no fuss