Phantom power is a small amount of voltage that is supplied to a condenser type of microphone that powers the microphone.
Condenser microphones are a type of microphone that use a thin membrane that is positioned very close to a solid metal plate. This membrane needs to be electricaly conductive. When sound hits the membrane, the capacitance between the plate and the membrane changes relative to the sound waves hitting it. These changes are then converted into a fluxuating electrical signal. However, this output signal bears very little current and is not audible when listening directly. This is where phantom power comes in. It provides the necessary current to boost the signal to an audible level.
Phantom power typically is primarily supplied in 48V, but also 24V and 12V.
Whenever you have an XLR output of a condenser microphone, it will most likely require phantom power to function. Some microphones, like the 3Dio FS XLR or 3Dio FS Pro II, also contain internal batteries as an alternate power source. This can be especially useful when recording outdoors or when a plug-in wall adapter is not available.