The atomic weight of an atom is the ratio of the atom's mass to 1/12th mass of one neutral atom of 12C in the ground state. Since the atomic weight is a ratio it has no dimensions. There is some controversy over the use of the expression atomic weight (since weight implies a unit). For an element, the elemental atomic weight is the weighted average of the atomic weights of all the naturally occurring isotopes of the element.
What's the difference between atomic weight and atomic mass?
Closely related to the atomic weight is the atomic mass. The atomic mass is defined such that the mass of a 12C atom is 12u. It follows that the atomic mass of an atom or nuclide measured in atomic mass units is numerically equal to the atom's atomic weight.
Example: Consider the nuclide Mg-26.
The atomic weight of Mg-26 = 25.982592929 (dimensionless)
The atomic mass of Mg-26 = 25.982592929 u
Naturally occurring elements usually consist of two or more isotopes. Magnesium, for example, has three stable isotopes - Mg-24, Mg-25, Mg-26. The isotopic abundance of an isotope is the fraction of the atoms in the element of that isotope. For an element, the elemental atomic weight is the weighted average of the atomic weights of all naturally occurring isotopes of the element, weighted by the isotopic abundance for each isotope.
Notice also that the average atomic mass is the weighted average of the atomic masses of all naturally occurring isotopes of the element, weighted by the isotopic abundance for each isotope.
Fubndamentals of Nuclear Science and Engineering, J.K. Shultis, R.E. Faw, 2002.