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There are a couple of main types of microphones - dynamic and condenser.

Dynamic microphones use a moving coil attached to a diaphragm.  When sound reaches the microphone, the diaphragm vibrates, causing the coil (made of metal wire) to vibrate as well.  There's a magnet by the coil, and as the metal coil moves by the magnet, it creates an electrical signal in the coil that becomes the signal to be amplified.

A condenser mic has a diaphragm mounted in front of a charged back plate.  As the diaphragm moves, this creates differences in the capacitance of the plate, which is what's used as the signal.  For this reason, they're also called capacitance microphones.

Adding some (very) basic and generalized characteristic info to Noteboat's post:

Dynamics typically can deal with higher level sound pressure levels (SPLs), are more durable and typically cost less than condenser mics. Most performance workhorses such as the Shure SM57/58 and Bullet mics are dymanics. These are also known to add substantial coloring to the source audio -- a good or bad thing depending on personal prefs and application.

Condensers often are categorized as large and small diaphram -- each with its own characteristics in terms of noise, SPL, freq response and etc. Condensers typically provide more sonic detail than dynamics -- often great for acoustic guitar and nuanced vocals. Condensers also require a power source if not self-equipped with a battery. This is reason for that phantom power switch on mixers and multitrack units.

In the studio, it's not uncommon to use condensers and and dynamics together to mic various instruments. Of course, mic deployment is one of the studio engineering black arts.

It certainly is a black art!

What the studio alchemists tell me lately is that condensers seem to be better suited to all-digital studios -- capturing every nuance lets them choose what to keep, what to alter, in a way that you can't with a dynamic source.

Dynamic mics aren't going to die out anytime soon, though - as he says, they're workhorses, and you'll hardly ever see a live act without a forest of dynamics in the setup.

condensers seem to be better suited to all-digital studios -- capturing every nuance lets them choose what to keep, what to alter, in a way that you can't with a dynamic source.

I would have to agree.  However, I do use both in my home studio.  Different situations call for different mics.  

I prefer to use condensers, they seem to give a warmer tone,  if you have a good one.  In my opinion, dynamic mics give you more of a generic sound,  no character.