The particles in a plasma have different velocities in general.
The distribution in velocity is a clue to their history: a stable,
relaxed, thermal distribution will have a Maxwellian shape.
Completely unrelaxed independent populations will have a "bimodal"
distribution.
A "tail distribution" is an intermediate situation. In such a case
we have caught the medium in the common (really, obligatory)
process of relaxation (a.k.a.
thermalization or Maxwellianization).
The figure below shows these three cases graphically:
To characterize the "tail" distribution we normally take the
logarithm and make a linear fit to the slope of the hard X-rays:
this is the "power-law index".
A "softer" spectrum has a larger index and a "harder" spectrum
has a smaller index.
We also refer to harder powerlaws as "flatter" distributions
sometimes.