Assuming you have a single handle valve, Moen has 2 types of cartridges. The Posi-Temp (replacement cartridge # 1222) turns on and off simply by turning the handle left to right and right to left. The Moentrol (replacement cartridge #1225B) turns on by pulling on it. Handle style does not aid in identification.
Buy your new cartridge before you start and buy genuine Moen! OEM cartridges can give Moen a bad name.
Once the water to the valve is off, remove the handle.
The procedure is the same for either type of valve so use the link below as a reference.
Firmly grab the temperature limit stop and pull it off the valve body. Using pliers, pull out the brass retainer clip. Place the notched end of the white plastic tool that came with your new cartridge onto the old cartridge and rotate as needed until completely seated on cartridge. Put a wrench onto the squared end of the white tool and turn back and forth several times. Remove the tool, grab the brass center of the cartridge with pliers and pull the cartridge out.
Briefly turn the water on ond off to flush the valve of any possible lingering particulate.
Insert new cartridge and use white tool to assure cartridge is all the way in then use the squared end to rotate the cartridge as necessary into an alignment position to reinstall the retainer clip.