Is my understanding of the concept of a mole correct?
The mole is a simply a number ($6.023 \times 10^{23}$). 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.
The molar volume $V_\mathrm{m}$ is the volume of a substance per mole of substance.
This is different from the "volume of one mole of substance".
For example, the volume of one mole of water is approximately $18~\mathrm{cm^3}$. But the molar volume of water should be $18~\mathrm{cm^3~mol^{-1}}$.
Saying that the molar volume is the "volume of one mole of substance" is incorrect, just like saying that the speed is the "distance travelled in one second" is incorrect. The speed is the distance travelled per unit time and therefore has units of $\mathrm{m~s^{-1}}$. I have had this "argument" with others before and I will stand by what I say.
Furthermore, there are no restrictions on the temperature, pressure or other conditions whatsoever. There is no restriction on what the compound is so the other answer is absolutely wrong on this.
The mole is not a number but the name of an unit: the unit of amount out substance. Amount out substance is the number of elementary entities divided by the Avogadro constant.
IUPAC's related definitions are somewhat recurrent in my opinion.
We can say that the amount of substance of 12 grams of carbon-12 is $1\,mol$.
Molar mass of something is the value of the variable that represents its mass divided by the corresponding to its amount of substance.
Molar volume of something is the value of the variable that represents its volume divided by the corresponding to its amount of substance.
Example: if you have 24 $g$ of carbon-12, its mass is represented by the variable $m = 24 g$. Its amount of substance by $n= 2\,mol$, so the molar mass of your solid is: $m/n = 12\,g\cdot mol^{-1}$.
The mass of an amount of your solid which correspond to 1 $mol$ is 12 $g$, which is different to $12\,g\cdot mol^{-1}$.
You should check IUPAC Gold Book whenever you are in doubt about chemical related definitions.
A mole is defined as: 6.023 x 10^23 elementary units of "something." That something can be atoms, molecules etc. So yes, I think you got that right.
Molar volume is defined as the volume occupied by one mole of an ideal gas at STP (standard temperature and pressure) which is approximately 22.4 liters.
