What gives rise to the normal distribution? I'd like to know if anyone has a generally friendly explanation of why the normal distribution is an attractor of so many observed behaviors in their eventuality. I have a degree in math if you want to get technical, but I'd like to be able to explain to my grandma as well

To my mind the reason for the pre-eminence can at best be seen in what must be the most electrifying half page of prose in the scientific literature, where 
Clark Maxwell deduces the distribution law for the velocities of molecules
of an ideal gas (now known as the Maxwell-Boltzmann law), thus founding the discipline of statistical physics. This can be found in his
collected papers or, more accessibly, in Hawking's anthology "On the Shoulders of Giants".   The only assumptions he uses are that the density depends  on the magnitude of the velocity (and not on the direction) and that the components parallel to the axes are independent.  Mathematically, this means that the only
functions in three-dimensional space which have radial symmetry and split as a product of three functions of the individual variables are those which arise in
the normal distribution.
