We might start with a study of concentration on an atom per atom basis. Assume that for every 99 atoms of water we have 1 atom of an impurity.
If we convert atoms to moles for both water and the impurity, the concentration on a mole per mole basis will still be 99 to 1.
Unfortunately, no chemist is good enough to look at a solution and see the number of atoms of water and the number of atoms of impurity. And no chemist can see moles either.
So we go to an Erlenmeyer Flask, which has a line and we know if we fill it to this line, we have the specified volume, something written on the Erlenmeyer Flask.
We may calculate that to get A moles of some solid, we need B grams of that solid, so we add B grams of the solid to the Erlenmeyer Flask and then we add water, swirling the mixture to dissolve B into the water, and we are careful to stop adding water when reach that line on the Erlenmeyer Flask. This is how we can get a concentration of moles/volume or grams/volume. From our knowledge of Chemistry we can calculate grams from moles, and vice versa.