A few more details, because this topic is close to my heart.
First a definition of terms:
Peak Level = the volume of the highest peak of the wave form
Loudness = the perceived volume of the track to a human listener. Two tracks can have the same peak level (i.e. they are both ‘normalized’ to peak level) but they can have drastically different loudness, caused by different amounts of dynamic compression.
We are aiming to compensate the difference in loudness between tracks, purely by adjusting the playback level of the whole track.
(Theoretically you could achieve the same by compressing the lower loudness tracks more. But this method is just purely evil and will therefore not be pursued. If you want to listen to the results of that approach, get a radio.)
The drastic level difference between the waveforms from VLC and Plexamp in the above picture Level/dynamic issue is easily explained:
The test track used is a very highly compressed one. Which means it has a very high loudness. Plexamp duly reduced its playback volume rather drastically, so other tracks with less compression have a chance to be perceived as equally loud.
Why not rather raise the volume of the quieter track, you ask?
Because then, the peaks of the quieter track would exceed what the digital-analog converter (DAC) of your computer could handle.
There is a technical limit to what voltage a DAC can produce in the peaks. If you raise the level of a track too much, you get distortion. (the peaks of the waveform get clipped at the highest level of the DAC. This is commonly known as ‘clipping’.)
One user proposed an approach like in other players, which do feature an adjustable preAmp value. Raising this preamp value would result in distortion, we don’t want that.
Some players have a ReplayGain mode named ‘prevent clipping’. Why not use this?
Because what these players do is employing a ‘peak limiter’, which reduces the level of peaks of the music. It’s effect is the same as a (very poorly executed) dynamic compression. We don’t want that, so it is not used.
As a user of Plexamp, you can get a very intuitive visual representation of the loudness of a track. It conveys a much more detailed information about a track’s loudness than the above mentioned DRx value.
Simply switch to the ‘Soundprint’ visualizer of Plexamp.
It shows you the average loudness of ~ 50 segments of a track. The longer a bar extends from the center, the louder this segment is.
Here is an example of the result of the ‘loudness war’, illustrated with the Soundprint viz:
The Beatles - Sgt. Peppers Lonely Heartclub Band (Remaster from 2009):

The same track from the ‘50th anniversary Deluxe edition’ from 2017:

As you can see, there are more longer bars. Which means the sound was compressed even more than on the 2009 release. The quieter passages were drastically raised in volume, reducing the dynamic of the recording.
For the technically inclined: Plex server determines the loudness of a track with the current state-of-the-art EBU R128 algorithm.