Condenser mics can be extremely high quality. The diaphragm can be very light, rendering a flat frequency response (with a small resonance peak at above 12kHz). Output of condenser mics is much higher than for dynamic mics. High output makes it more robust to noise. To charge the capacitor a source of power is needed (usually phantom power). An alternative to using a power source is to introduce a permanent electrostatic charge during manufacture, resulting on the “electret” mic. Electret microphones can be very small, high quality (back electrets) and cheap, e.g. Tie-clip TV microphones
How do they work?
A capacitor has two plates with a voltage between them. In the condenser mic, one of these plates is made of very light material and acts as the diaphragm. The diaphragm vibrates when struck by sound waves, changing the distance between the two plates and therefore changing the capacitance. Specifically, when the plates are closer together, capacitance increases and a charge current occurs. When the plates are further apart, capacitance decreases and a discharge current occurs. A voltage is required across the capacitor for this to work. This voltage is supplied either by a battery in the mic or by external phantom power.
Where are they used?
Condensers are more studio recording. They're not really meant to be carried out on stage like a Dynamic microphone would be. These types of microphones are good for voice overs, podcasts, singing and acoustic recording.