The quick and dirty explanation: dynamic mics (the type you usually see for school assemblies, outdoor concerts, and such) function by using a barrier with a magnet attached. The barrier/magnet (generally referred to as a diaphragm) is next to a coil of wire. The movement of the magnet with a wire nearby produces a small electric current. This current is fed into a "preamp", which makes that tiny signal much louder (bumps it up to what is called "line-level input"), and then you can record or hear it.
Ribbon microphones function by a similar principle. However, rather than a robust circular diaphragm, they use a small ribbon with a magnet on either end. This makes them more sensitive to sound than a traditional dynamic mic, with some pleasant natural compression, but they are also extremely delicate and expensive. That ribbon can break easily. However, they produce very rich vocals, particularly for female vocalists. Generally, it takes much less air pressure to produce motion in the relatively large, flat ribbon. However, the ribbon is also quite sensitive to location; you'll have a steep decibel dropoff with a singer singing at the side of a ribbon mic.
Now, so far, these microphones require no power to run them. They generate their own current through the movement of the sound waves you've created.
Enter the "condenser" microphone.
Essentially, the diaphragm in a condenser microphone is a capacitor. That is, it's an electrical apparatus which holds a charge. One plate of the capacitor vibrates in response to sound waves. The second plate is steady. The capacitor's capacitance (farads) changes depending on how far apart the plates are. The plates in most decent quality condenser mics are gold, in order to resist corrosion and provide a longer useful lifetime for the microphone. (Well, they're actually gold dusted onto a plastic diaphragm). Some of the cheaper ones, like the ones powered by AA batteries that you buy at Radio Shack, use steel or silver instead of gold, and are only useful for a few years before their frequency reponse diminishes.
Savvy so far? A capacitor stores electricity. The amount it can store varies based upon the materials used for the two capacitor plates and the dielectric (the stuff between the metal plates). In the case of microphones, the "dielectric" is just air between the two nearby plates.
The net result of measuring a capacitor's capacitance changes, versus the small electric current changes created by a moving magnet in a dynamic mic, is that the condenser microphone is far more sensitive to smaller and shorter sound waves. In other words, you can play softer sounds and they'll be picked up. You can also play higher-freqency sounds which a dynamic mic misses because the sound wave isn't large or long enough to move the magnet noticeably.
In the case of the condenser mic, the electric charge to the capacitor is provided by a "phantom power" source. Without phantom power, condenser mics simply don't work, because their capacitor has nothing to charge it. Phantom power is provided right along with the regular signal. It is extremely small, and phase-cancelled if your input is balanced.
Phantom power is only needed for condenser microphones. Turning on phantom power will not harm dynamic or ribbon microphones. Usually, you will want to turn on your mixer, turn on phantom power, and THEN turn up your master volume level if you value your ears.