MM, MC, or MI cartridge

MC cartridges are generally considered the better sounding option because of their lower moving mass. They are better at picking up details in the record grooves and can therefore reproduce the fine details in the music with higher accuracy than MM cartridges. The differences between MM and MC are more apparent when listening to very high quality recordings on a high-end turntable, a good quality MC cartridge like Hana ML will produce more precise soundstage and finer details than a similarly price MM cartridge.

This is not to say that MM cart can’t sound good. It is my opinion and understanding that MM cartridges with their higher moving mass, have a tendency to “smooth out” the fine details. For affordable low-end turntables, MM cart is the most obvious choice :-)

And then there is difference of signal output levels. The typical signal output from a MM cartridge is 5mV. While a MC cartridge typically outputs 0.2mV.

Generalities are dangerous.

Not an easy answer, the main thing is whether an arm can properly track a cartridge, this is the most important.

Then you have loading, differernt for MC's and MM/MI. Any type should be loaded correctly.

It all depends on how the cartridge is voiced.  The different types do have slightly different characteristics, but a MM can be really warm or very bright.  The same holds true for other types of cartridges.  Some are very neutral, while others are voiced to be more euphonic.  

I prefer MC cartridges, as they do tend to present more details and have less "grain", however I'm currently enjoying my Soundsmith Aida II MI cartridge, which comes very close to the performance of my much more expensive MC cartridges.

Most MC carts require more gain due to lower output.  They are not all "typically" .2mV.  Many are in the .4 - 5mV range.  There are high output MC cartridges that don't require as much gain (usually around 2.0 mV) and may work with a MM phono stage, but typically don't sound quite as good as their low output versions.  

It's important to match the cartridge to your tonearm and phono stage.  A well matched MM may sound better than a similarly priced MC if not matched well to the rest of your analog chain and your listening tastes.

In other words, MC, MI, MM is just one factor to consider when purchasing a cartridge.