The frequency response of a microphone is determined by its ability to translate different frequencies at different volumes consistently. A microphone with a flat frequency response would translate any given frequency at any given pressure level accurately.
Some recording and instrumentation microphones get as close as they can to a truly flat frequency response, but some microphones are designed to have emphasis in a certain frequency range. These microphones have gains or reductions of certain frequencies to achieve advantage where applicable. This is useful when you want to make up for a deficiency in the sound source.
Lower end microphones tend to have a frequency drop off around 100Hz. This works well when capturing vocal because the human vocal range rarely goes below 100hz, so with the dropout, you will be eliminating extra noise. For instrument recording, a drop off at around 50Hz is workable. Many microphones do have a raised response in the higher frequencies; this is called a presence peak and is usually a characteristic of vocal microphones.
With directional microphones, it is important that the frequency response remains consistent as the sound hits off axis, even though the sensitivity will drop. If the frequency response does change as the microphone becomes of axis, the tone of the sound source will change. The most susceptible microphones are hand held microphones as the axis will continually shift when in use. Even if the microphone is stationary, the frequency response should be the same even off axis so the reverberant energy will not be consistent with the source sound
A consistent frequency response is a characteristic of a quality vocal microphone, and is much more important than the overall sensitivity, or the response on axis. If the frequency response becomes rough off axis, then the tonal quality of the vocalist’s voice will change as they move around.
Frequency response is a huge factor when selecting a microphone for a specific application, and this is best done by ear. If a microphone has a peak between 5Khz and 8Khz, then that microphone is better suited as a vocal or solo instrument microphone. A microphone with a pronounced low end and less high end response would be suitable for drums or bass.
No two microphones sound the same and no two instruments sound the same, so you can’t rely on the spec sheet to give you the answer, you need to trust your ears!
(Example of a vocal microphone frequency response)