This is the very common Moen Legend/Chateau. The cartridge is removed by popping out the bronze clip....you can just see the top of the clip sticking up, has a tiny hole in it. Turn off those hot and cold stops first, and be ready to turn off the main if you pull the cartridge and the stops dont hold.
In theory, Moen cartridges are an easy repair job. In practice, they CAN be very difficult if the cartridge is stuck, with water deposits, like it appears likely in your case. You will need to purchase a Moen cartridge removal tool from the hardware store. And don't do this on Sunday morning! Do it Wednesday morning, so after you get stalled, you will still have time to call in a plumber and not go all weekend with NO water.