Use an adjustable wrench to unscrew the locking nut holding the faucet cartridge in place and use pliers, if necessary, to pull the cartridge out. An Allen wrench or Phillips Head screwdriver may be necessary to remove the faucet handles in order to access the cartridge.
Before removing the faucet handles and cartridge, make sure to find the water shutoff valve for the faucet to turn the water off. Allow the faucet to drain any remaining water before starting the repair. When removing the cartridge, note the location of the notches in the faucet to make reinstalling easier. If pliers are necessary, wrap the cartridge with tape or a rag in order to protect it. Once the cartridge is removed, it may be helpful to bring the part to the hardware store when buying a replacement.
Before replacing the faucet cartridge, remove the old seat and spring using a small screwdriver. Place the new spring into the gap, then push in the new seat. Rub plumber's grease on the base of the new faucet cartridge and push it into the faucet. Make sure that the new cartridge aligns properly with the notches in the faucet. Screw on the retaining nut and put the faucet handles back in place.