Use the noun atomic mass when you're talking about the mass of an atom's protons, neutrons, and electrons.
You'll come across the term atomic mass if you study chemistry, in which it refers to a single atom of some chemical and the total mass of that atom. The unit of measurement is an "atomic mass unit" or amu. Mass is a measurement in science which is essentially the same as weight, without taking changes in gravity into account, and atomic means "of an atom." The two were first combined into atomic mass in the mid-1800s.