Turn off the water supply at the mains. Look around for the main water turn-off--it may be outside.
Open the tap you are going to work on and allow any residual water to drain.
Flip out any plastic cover on the tap handle. Use a screwdriver or knife to pry it off. Unscrew any screws you see.
Remove the head. Wrench it off. You need to be careful here and maybe add some padding if the head is chrome, as the wrench might damage the chrome.
Look for the washer and remove it or the broken pieces.
Replace the washer. Match it up from your kit.
Replace the head and any screws or plastic covers. Close the tap.
Turn the water back on at the main inlet.
Go back the work site and open the tap. Allow the water to flow for a bit.
Close the tap and inspect for leaks.
Repeat changing the washer slightly if you get any drips.