Phono cartridges are transducers that convert mechanical movement into electrical energy. The other transducer in your audio system is your speakers, which operate in the reverse fashion.
In a nutshell, the main difference between moving coil (MC) and moving magnet (MM) cartridges lies in the construction of the transducer assembly. The three main elements in the transducer assembly are the stylus/cantilever, the coil, and the magnet. The electrical engineering types can give you a more technically sophisticated answer, but here is the simple version.
MC cartridges have the coil wound around the end of the cantilever, and this coil moves within the magnet's field to generate a voltage signal. Because the coil is wrapped around the end of the cantilever, there isn't a lot of room for numerous, heavy wraps. There is a plus and a minus with this approach: the plus is that the low mass of the coil allows it to move very quickly as the stylus traces the LP's grooves. The minus is that the small number of coil windings results in a low voltage output, which means that the signal must often go through a step-up transformer or "pre-preamp" before going to the main preamp. The theoretical advantage of the MC cartridge is greater transient speed, and sometimes more extended high frequencies.
Moving magnet (MM) cartridges essentially have the coil and magnet assembly reversed from that of the MC. The magnet structure is located at the end of the cantilever, and it moves within a coil assembly. Since the coil can be larger (i.e., have more windings), the signal generated by the movement of the cantilever is higher. MM cartridges, therefore, typically generate a much larger voltage than MC's, which negates the need for a step-up device before the main preamp. The minus, if there is one, is slower transient speed, and (maybe) less extended high frequency response.
Having said all of that, the relative advantages and disadvantages of each cartridge type tend to be less noticeable with lower-priced models. The Shure V15VxMR you mention is a fine cartridge that performs very well at its price range. Like many audiophiles, I have owned a number of Shure V15's over the years, and recommend them without hesitation to someone who wants a lot of performance for the money.
There are other options, however, that you may want to consider. The Grado line of cartridges also offers a lot, particularly in its lower-priced models (in the $100-300 range). The Grados are usually classified as a "moving iron" cartridge, but they operate like a MM, and usually have an output signal comparable to the MM cartridge.
You will often get a lot of spirited debate between the MC vs. MM cartridges advocates, but except for the really pricey models ($1000 and up), the sonic differences are fairly subtle. At the lower priced range you seem to be considering, the preamp will be as important to the eventual audio quality as the cartridge itself.