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Dynamic, Condenser, and Boundary
Most Mic's you will find in the recording studio fit into these categories.
The dynamic and condenser are the most used of the 3.
To simplify all this, a condenser Mic will have a more accurate sound so they will work well on acoustic instruments and vocals. These Mic's need phantom powering and have a hotter signal. Because of this they tend to not work so well on very loud instruments such as a guitar amp or close miked drum.
At higher volumes they will sound more brittle or even distort.
These Mic's tend to sound very good so they are a mainstay in the studio. Since they are so sensitive you wont see them much on stage or in live performance.
Dynamic Mic's are the next most popular Mic in the studio. They can take loud volumes and still sound good. This makes them the best choices for close miked drums, guitar amps or very loud singers.
Most Mic's used in the studio these days fit into these categories. Another Mic you may have heard of is the boundary Mic.
The boundary Mic is also a condenser Mic where the capsule is mounted in the center of a flat metal plate. Sound is reflected off this plate so it reaches the capsule at the same time.These Mic have a tendency to pick up large areas as they are omni directional.
I have seen these Mic's used in small studios where they don't have tons of Mic's to record a drum kit. Two of these Mic's can pick up most of the drums very evenly with a relatively good sound. With a dynamic Mic on the kick you get a budget way to record drums that can be surprisingly effective.