There are two things that routinely mystify science students:
anything to do with amount of substance (now to be called "chemical amount"), the mole, and the Avogadro constant (or the Avogadro number), and
anything to do with the now-you-see-me-now-you-don't radian. Let me address the first.
If we have a general number of entities of kind X (e.g. X is the chemical symbol) represented by N(X), the corresponding chemical amount of X is denoted by n(X), which is an aggregate of N(X) entities. In symbols:
n(X) = N(X) ent, where ent represents an amount of one entity (atom, molecule, ion, sub-atomic particle, . . .), i.e. the entity itself.
The Avogadro number is the (dimensionless) ratio of one gram to one "atomic mass unit" (now called dalton, Da): g/Da. One mole is an an Avogadro number of entities: mol = (g/Da) ent. Thus we have the important relationship: Da/ent = g/mol = kg/kmol, exactly. In other words, at the atomic level, the appropriate unit for amount-specific mass ("molar" mass) is dalton per entity--and, because of the mole definition as an Avogadro number of entities, dalton per entity is exactly equal to the macroscopic units gram per mole or kilogram per kilomole.
The critical problem is that IUPAC does not have a recognised symbol for one entity. It is sometimes (incorrectly) thought of as the (dimensionless) number one. In which case the "mole" is simply another name for the Avogadro number: "mol = g/Da". In this case we have the (incorrect) relationship: "Da = g/mol". Tables of "atomic weights" list the numerical values of atomic-scale masses in daltons--e.g. Ar(O) = ma(O)/Da = 16. The corresponding amount-specific mass is M(O) = 16 Da/ent; and this is (exactly) equal to 16 g/mol or 16 kg/kmol.