Just a clarification, I’m assuming that you’re asking about dynamic compression, which affects signal levels, not data compression such as mp3 encoding.
Compression changes the output level of a signal based on the input level and a handful of user settings. The most important two are _threshold_ and _ratio_. The threshold is the level at which the compressor starts modifying the signal. Signals below this threshold pass from the input to the output without any change applied. Signals above it are then reduced in level by the chosen ratio. For example, a 4:1 ratio will require a 4db increase in input level _above the threshold setting_ to output a level that is 1db above the threshold. The increase in volume was “compressed” 4x. A “limiter” is a compressor with a very high ratio, from 10:1 – infinity:1, these are used when you don’t want the signal level to get _any_ greater than the threshold, to avoid overloading the next thing in the chain perhaps.
In terms of how it sounds, it can range from unnoticable to a very pronounded “pumping and breathing” effect which, while hard to adequately describe, is very obvious once you’ve heard it in action. Some uses include setting a low threshold and a small ratio, then using the final make-up gain on the compressor to increase the level by the amount compressed. This has the effect of bringing quieter parts of the performance more to the forefront, while keeping the louder parts from getting too loud. Another use is for effect, this is often done with _parallel compression_ where a whole mix/submix (say the drums in a pop mix) is compressed at a quite high ratio in order to get the distinctive over-compression sound, and then that output signal is taken and mixed in at a lower level than the uncompressed signal. This effect adds the feeling of energy to a mix, without just pushing volumes until things get out of control.
Many individual instruments use compression as part of their signal chains, too, such as electric bass or guitar. Their pedalboards often have compressor pedals, and they tend to use them to increase the sustain in the sound, or to help even out otherwise uneven levels from techniques like slapping the bass.
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