Phantom Power is a term given to the process of delivering DC (Direct Current) to microphones requiring electric power to drive active circuitry. Condenser microphones such as Shure's KSM range all have active circuitry and require phantom power.
 

The power can be provided by a battery located inside of the mic; an example is the Shure PG81 (now discontinued) that operates from a single AA battery. Alternatively (and most commonly) the DC power is provided by the pre-amp/mixer and delivered to the condenser microphone via the mic cable. This method is referred to as phantom power. The worldwide standard for phantom power is 11 to 52 volts of DC (typical studio mics run on 48v). Your preamp will typically have a button labelled 48v, which allows you to turn this on/off. However, some older mixers and cheaper audio interfaces may not have phantom power. In this case, an external phantom power supply can be added between the condenser mic and the preamp.
 

A dynamic microphone, like the SM58, does not require phantom power because it does not have active electronics inside. Nonetheless, applying phantom power will not damage other microphones in the vast majority of cases. The reason is that modern dynamic  microphones are designed to accept phantom power without issues, but we advise checking your manual or consulting with the manufacturer first before connecting; particularly if you have a ribbon microphone.

Additionally, it's a good idea to turn phantom power off while plugging and unplugging microphones to prevent any potential power surge and general pops and loud noises, which could damage your speakers/headphones over time.

Condenser microphones made in the 1930s, 1940s, and 1950s required a special power supply to operate. This power supply would often be located quite near the microphone and was usually large, heavy, and cumbersome. In the 1960s, work began on a new powering concept that would eliminate the need for a separate power supply.

Schoeps and Neumann (German microphone manufacturers) were leaders in this development. Eventually, a new condenser mic powering standard emerged. The DC power to operate the condenser mic was provided by the mixing board and delivered via the mic cable; eliminating the need for an external power supply. And what does one call a power supply that is working, but invisible? It is a phantom power supply!