When one refers to molar mass, they might as well be saying "molecular mass". This mass may be calculated by referring to the periodic table of elements and looking at each element's mass number. Add all the atom's mass numbers together, and you get what is referred to as mole mass, or the old terminology, molecular mass. This type of mass is usually referred to in units of "grams per mole", meaning "grams per molecular formula"
The mass in grams of a compound would be a specified amount of that compound, measured out on a triple-beam balance, and electronic scale, or some other type of measurement device designed to measure mass. You are measuring an assigned quantity of that compound, usually in grams. It is an actual physical, quantitative measurement of a substance you can see, much larger than the atoms that make the substance up.