there are at least two different meanings of "compressed audio", one is, as your title mentions, compression of data and the other is level compression. they're two totally different functions.
about compression of audio data, there are two general classes: lossless compression and lossy compression. in a crude sense of the word, signal quantization is a form of lossy compression, but i would not call word-width reduction the same topic as compression. so let's assume the audio data has already had its word-width (or "bit depth") reduced to its production length (say, 16 or 24 bits or whatever).
as the names might suggest, losslessly compressed audio data will have the audio data restored exactly when uncompressed. the original waveform will be restored. for lossy compression, when the audio is uncompressed the original waveform is not restored, but the expectation is that the audio will sound the same.
to do lossless compression of audio, there are two main steps, the first is what is commonly called "Differential Coding". "Delta Modulation" is the simplest example, but one can employ Linear Predictive Coding (LPC) to combine the past N samples in such a way as to predict the next incoming sample. if the prediction is good, the difference between the actual sample and the predicted sample will be small and that reduces the number of values to encode significantly. say, for 16-bit, instead of having 65536 values to encode, your difference signal might be not much bigger than 128 and you'll need only 8 bits to encode that difference.
what about the rare times you need more than 8 bits because the difference is greater than 128? this is where entropy coding or Huffman coding comes in. this is essentially what file compression programs like pkzip do. words that appear often are assigned fewer bits than words that appear rarely. this all comes out of Shannon's Information Theory.
that's, in a short form, what lossless compression is about. to learn what lossy compression is about, there is a whole additional science about perception of audio, what audio features can be heard and what are not heard. it's hard to explain, but there are ways of separating what can be heard from what cannot and for the separated data that is deemed not audible, no bits are allocated to it. it's a crude explanation.