Here is a sequence of 5 short generated sounds, separated by 1/2 second silence.
Assuming that you listen on half decent speakers or better, I expect that you would agree that they all sound roughly the same loudness. (Laptop speakers fall down badly on this test because they have such limited, and non-linear frequency response). I’m not suggesting that they are exactly the same loudness - just the same ballpark. Each of the 5 sounds has been individually normalized to -12 dB rms.
This second sequence is the same sounds, but in this case each sound was normalized to -3 dB peak.
Notice in particular that the final sound is much louder than the noise that precedes it.
Whether producing an audiobook, a podcast, a compilation album, a radio show,… there are many cases where you may want multiple different tracks to be around the same loudness. Peak level is often a very poor measure of loudness. RMS level is certainly not perfect, but it generally gives a much better indication of loudness than peak.
There are better ways to measure loudness, such as LKFS, ReplayGain, iTunNORM, and many others. Advantages of RMS measurement are that everyone in the audio business understands it, it’s been clearly defined as a scientific measure for aeons, and it is simple to measure.
Let’s say that I’m making a podcast and that you are making a short article to go into it, but it’s all very last minute so I want you to send me a recording that I can just drop into the show and run it. Obviously I will need the peak level to be below 0 dB (and allow a bit of headroom for transcoding), and I’ll want it to be about the same loudness as the rest of the show. If the rest of the show is around -17 dB rms. and your article is around -17 dB rms, then we’re probably close enough.