There is a common misconception that the Avogadro constant is one of the fundamental constants of nature, in the same category as the speed of light, the Planck constant and the invariant masses of atomic-scale particles. Although the absolute mass of any specified atomic-scale entity is an invariant universal constant of nature, the Avogadro constant relating this to a macroscopic quantity is not. Rather, it is a man-made construct, designed by convention to define a convenient unit relating the atomic and macroscopic scales. The misportrayal seems to stem from the widespread use of the term 'fixed-Avogadro-constant' for describing a redefinition of the kilogram that is, in fact, based on a fixed atomic-scale particle mass. This paper endeavours to clarify the role of the Avogadro constant in current definitions of SI units for mass and amount of substance as well as recently proposed redefinitions of these units—in particular, those based on fixing the numerical values of the Planck and Avogadro constants, respectively. Precise definitions lead naturally to a rational, straightforward and intuitively obvious construction of appropriate (exactly defined) atomic-scale units for these quantities. And this, in turn, suggests a direct and easily comprehended two-part statement of the fixed-Planck-constant kilogram definition involving a well-understood and physically meaningful de Broglie-Compton frequency.