Moving Coil Cartridges are called named so because the fixed magnets on them are attached to the cartridge body while the wound-coil electro-magnets are attached to the moving stylus. The wound coils move as the stylus goes up/down tracking the groove, so we call these Moving Coil cartridges.
Moving Coils generally have lighter stylus assemblies so the stylus can move up/down to the music faster and with less inertia. This gives better music timing and more information we can hear. Moving Coils generally have low output electrical signals of .25 mV to .7 mV as compared to the 2.5 mV to 7 mV of Moving Magnet cartridges. Because of this lower output, higher built specification phono stages need to be used to amplify these smaller electrical signals without introducing noise into the sound. That's why MC phono stages are so much more expensive.