RMS or Root Mean Squared is the Root of the Mean of the Square of each sampled value of a signal (for discrete values) or the Root of the Definite Integral of the Square of each bounded function (for continuous values). 

     But what exactly does it mean? 

     In my opinion, it is a means of central tendency (I use the word "like" because it doesn't imply the same connotations as an average), only that negative integers are treated as positive values by squaring them in the first place. After squaring, the "mean" is taken, then the square root to reverse initial squaring. 

     Thus, the entire signal is sufficiently represented by a single constant value. This constant value can be treated as DC (direct current) since DC is constant unchanging voltage. When we take an RMS measurement of an AC signal, we are actually getting an equivalent DC signal that can replace that AC signal in terms of power delivery.

      But why not use Cube-Root Mean Cube (CMC) or higher exponentials? When observing change in  final value, sensitivity to change reaches the highest value when squared and gets worse when you "CMC" or treat it with higher exponentials. 

     For example, take the vector [10 20 12 15 7 9]. Its RMS, CMC, etc. is 14.6002, 15.2561, and 15.8017 respectively. When you try and tweak one component of the vector, say increase the first element by 3, yielding [13 20 12 15 7 9], the resulting RMS, CMC, etc. is 14.9889, 15.5366, and 15.9942. Getting the sensitivity of each:


RMS


14.9889-14.6002=0.3887


CMC (Cube-root Mean Cube)


15.5366-15.2561=0.2805


Etc. (raising each to the 4th , getting the mean, and taking the 4th root)

15.9942-15.8017=0.1925


     It is easy to see that RMS is most sensitive to change. Also, CMC is disqualified from the beginning since it doesn't remove negative components at all.