Mass is based up the 1kg (definition) bar in Paris.
As an SI base unit it's one of the fundamental dimensions which the other units/dimensions are based off of. For example Force is a function of mass and acceleration which in turn is mass, distance and time.
I always thought Avogadro's number was the number of species in 12 grams of carbon-12
They (COPM) are meant to be looking at how to redefine the kilogram (and hence mass).
edit: the big bang theory would suggest that energy turned into mass so it could be tenuously linked using distance, time and energy. Of course if the Higgs Boson is seen then that could be used to help define mass a little less abitrarily