"Ok, that's how microphones work, but what's the difference?" you ask.
You can't say something's categorically true of any specific kind of microphone, because there's just so much variety out there, and so many manufacturers out there doing new and innovative things. However, there are useful generalizations.
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Dynamic microphones are most of what you see, day to day. Compared to condenser microphones, they're "less crisp", which means less accurate and that they have worse high frequency response. This is due to the mass of the diaphragm, and that the sound wave must do work on the coil to cause the current. Much like cone speakers, the frequency response varies with size. A mic designed for a kick drum will have a diaphragm of an inch and a half, or more, whereas one designed for guitar micing or vocals will have one of an inch or less.
Dynamics are also the most resilient mics. That old adage, "You can hammer a nail with a '57" is actually true. I wouldn't recommend it, but you've got to try hard to kill one. Also, most dynamics can handle pretty high SPL, so you won't destroy one by screaming into it, or placing it next to a snare drum.
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Condenser Microphones have a much wider frequency response curve than dynamics, due to their lighter and more sensitive transducer. Their sound is also described as being more detailed. Their really high sensitivity is why they're useful for micing things from far away; if you've ever seen mics hanging down above a concert hall or theater, you can bet that they're condenser mics (or electret, maybe, but that's beside the point). There are varying diameters of condenser mics, small diaphragm condensers are used for micing instruments, drum overheads, etc, while large diaphragm condensers are used for vocals primarily. Small diaphragms are more detailed, while large diaphragms smooth things out a little, and have somewhat better low-midrange response (good for vox).
Condensers, though, are fragile. You can break one by saying "POP!" into it loudly. This is why you see singers in studios singing into a mic with a big circular thing between their mouth and the mic. That thing's (appropriately enough) called a Pop Filter, and it blocks the plosives by only allowing the sound to pass through, while blocking the blast of air. As Littlerubberfeet points out, though, what's more dangerous is moisture. Once some condensation gets in there, the mic is done.
I just want to air one of my pet peeves, here: most all condenser microphones made for vocals (large diaphragm) are side-address; you're supposed to talk into the side of the mic, not the end (sidetalking!). However, in innumerable things I'm exposed to on TV, or in movies, people talk into the ends of such mics! It's wrong, and it looks wrong. Usually, sound is one of the few technical things they don't get wrong in film, because there's an expert on set, but he (note the stereotype! Unfortunately, it's usually true...) must've been drunk or sleeping, or something... Anyway, back to our regularly scheduled programming.
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Exotic microphones. With the last two types of microphone (or even just the first!), one could make a hit album. Most things you hear on a CD have passed through one of the last two kinds of microphone, but there are a few more types, so I'll give them some mention here.
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Ribbon Mics are a very old design, and are seen as an anachronism, most of the time. They have detailed response like a condenser, but are dynamic. However, they are very fragile, and cannot handle loud sounds. Many have a figure-8 pickup pattern, leading to doing some interesting things with them, like setting up guitar cabinets on either side of the mic. There are some VERY good ribbon mics out there, and because they're obscure, they're generally used by people who've sought them out, and know what they're doing.
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Electret Mics are much maligned, since old designs really did sound horrible. Modern electrets are getting much better, though, starting to compete with condenser mics. They still have a stigma, however. Give one a listen if you're in a position to, they're not bad (anymore)!
Anyway, so that's a whirlwind tour of the wild world of microphones. This is only a rough outline, and full of half-truths for the sake of simplicity. If you really want to know, look it up! I recommend www.prosoundweb.com for articles and info. Feel free to /msg me if you have additions/suggestions/comments, etc.
All info in this node from my experience, and from my copy of Modern Recording Techniques, fourth edition, by David Miles Huber, and Robert E. Runstein.