What is the difference between Shure sm57 dynamic mics and the non dynamic mics?

well, depends on what you use it for. SM57's are usually used for guitar cabs, toms, snare acoustic guitars. condensors and ribbons are usually used for vocals and such because they are clearer and louderbut they are also more delicate and such.
 
a dynamic mic is basically a speaker in reverse - you have a moving coil(diaphragm) with a couple of wires(which are attached to other electronic components) coming off of it...as sound waves hit the diaphragm, it moves back and forth and produces a signal that is then sent through the cable to the preamp

condenser mics work somewhat similarly, but not totally the same. condensers have a backplate behind the diaphragm - between the plate and the diaphragm there's basically a shitload of electrons bouncing around. as sound hits the diaphragm, it compresses the area between the diaphragm and backplate. this is where condenser mics get their name, and it's also why they're usually more sensitive than dynamics. the change in pressure across the diaphragm is then converted to an electrical signal, which runs from the mic to the preamp. due to the fact that the area between the diaphragm and backplate has an electrical charge, condensers also require phantom power to operate - usually 48v, but some cheap gear will give off as little as(i believe) 28v.

in addition, you also have your ribbon mics. these are similar to dynamics, in that they operate by having a moving element transfer vibrations through a wire and electronic components, and on to the preamp. big difference here is that instead of sensing vibrations with a round coil, the vibrations are picked up using a long, thin strip of some sort of ribbon material that is suspended within the mic and held in place at both ends. most ribbons don't require phantom power - some, in fact will get toasted by phantom power, while there's a few models out there that actually require it.

as far as what mic is best on what source, that's completely and totally subjective. dynamics usually end up on extremely loud sources and drum kits...not always because they sound better, but because they tend to be much more rugged. condensers are typically considered favorable on anything that's going to be very detailed or have a lot of high frequency content - vocals, any string instrument, piano, cymbals, etc.

ribbons work on damn near anything, as long as they aren't subject to "puffs" of air, like with loud vocals with no windscreen, or right in the hole of the kick drum...shit like that will usually blow out the ribbon.