The mole is a simply a number (6.023 x 10^23). It's just like what a dozen would be like for eggs, except that the mole applies to atoms and molecules. If we were to count the number of atoms in 12 grams of carbon-12, it would be a mole.

Molar mass is the mass of all the particles in a mole of substance. Molar volume is the volume occupied by all the particles in a mole of substance.