When a portion of a performance gets too loud in comparison to the rest of it, compression is used to tame loud parts over a certain threshold, a common control on many compressors. Attack settings allow more or less of the start of a note to get through before compression begins, and release controls how the compression effect ends. The ratio of input signal to compressor output controls the intensity of the effect.
Overall loudness of a mix can be manipulated with compressors. By taming extreme transients, the level of the rest of the audio can be raised, giving the impression of a louder performance. Many mastering plug-ins do just that as part of the processing package. Care must be taken as excessive compression can sound harsh and may change the mix relationship between instruments. If you do not expect to use a mastering service, doing a mix with a stereo compressor patched into your master outputs can help you avoid these problems.
Using a compressor on a single element of a recording, like a vocal or solo instrument, can cause that element to jump forward into a feature role. With equalization, compression is one of the first steps taken when a vocal sounds buried and an increase in volume brings it too far forward. Broadcasting frequently makes use of compressors as a way to reduce the volume of background sounds behind voiceovers. This is an effect called "ducking."
Many compressors have the capability of controlling one sound with the dynamics of another through use of a sidechain. A common example of this is running an electric bass through a compressor to which a kick drum is inputted into the sidechain. The performance of the bass and drum will then seem to match more closely as the compressor envelope of the drum is overlaid on the bass. Sidechains are also used in the ducking procedure described above.