To give you a quick and non-technical answer, here are the descriptions of how a moving magnet, moving coil, and moving iron cartridge operate and differ from each other:
Moving coil: the inner end of the stylus cantilever is wrapped with coils of wire (more coils = more signal voltage), and this coil-wrapped cantilever end moves within the magnetic field of a fixed set of magnets. Generally speaking, moving coils have faster transient response than MM's, since there is less mass involved, but the output signal is normally lower than a MM.
Moving magnet: the inner end of the stylus cantilever has small magnets attached to it, and this magnet assembly moves within the coil assembly.
Moving iron: Grado makes so-called moving iron cartridge designs, and here is the explanation of the design that they present on their web site:
PIVOTED FIXED AXIAL CANTILEVER STYLUS-GENERATOR
This design feature is basically quite simple. The "OTL" cantilever shaft is brought to a fixed axial pivot that supports the entire cantilever assembly. A miniature element attached to the end of the cantilever (the "iron") is allowed to move freely within the lines of flux of a stationary magnet and coil structure. This system of support is mechanically more accurate (more linear) than conventional "teeter-totter" designs that utilize an iron armature to balance the mass of the cantilever. The teeter-totter design is supported in the center with a rubber donut mounting.
The fixed axial design has a very low tip mass, which results in lower distortion, improved transient response, and longer record life. This proprietary design allows the implementation of the Flux-Bridger Generator System.
FLUX-BRIDGER GENERATOR SYSTEM
The Flux-Bridger design uses four separate magnetic gaps that the miniature element of the cantilever (the "iron") bridges. The miniature element moves between opposing flux gaps creating an increase in flux in one gap while reducing it in the other. The four separate magnetic gaps create a highly efficient and perfectly balanced system. This design requires fewer coil turns than conventional designs.
The Grado web site has a good diagram of the internal structure of their moving iron cartridge. Here's the link:
When you get the home page, there are three tabs in upper left corner -- click on "Cartridges".