That's a Moen cartridge. After shutting of the water to the shower you have to remove the brass clip that at the top front of the valve base. There's a small hole that some needle nosed pliers fits in just great, pull the clip straight up and out. Now you'll have to twist and pull the cartridge out of the valve base. You'll probably need the pliers again since you won't have the little removal tool unless you've done this before. Spray in a little CLR to break down the lime build up. It'll take some good twisting and pulling but eventually it will break free.

The new cartridges are readily available from your home store or plumbing supply store. There are similar ones so you'll need to bring the old one in to get a perfect match. The picture below shows what it should look like. When re installing the new valve, don't forget to put that brass clip back in.