How have I to understand in what those quantum numbers refer for ortho and para water? I need to make an English presentation at my School (Graduate school) about something people may don't know. I chose to speak about the separation of ortho and para water. I first chat here with @Orthocresol about questions I have but this is really tricky and I'm not understanding that so much. So I chose to ask the question here, it might be helpful for other people and I hope at least for me. 



At first my question is to understand what the notations $\lvert0_{00}0 \rangle$ and $\lvert1_{01}M \rangle$ in the article mention before about different states of water refers too? 

What are those four numbers if the ket notation, from where they come from and what does they mean? I know it makes a lot of questions but maybe are they linked? 



I read this article on wikipedia and this one too, and I also found that but I don't understood a lot of things. Maybe that $J$ is a quantum number which quantify the rotation energy of a molecule. But I have a lot of troubles with the projection and I don't know what $M$ is.



I would be glad if the answer was in a part like popularisation because I guess I'll have some troubles to understand it. But I have good skills in algebra, I had a twenty hours introduction in quantum chemistry, the hardest thing I did was a presentation of the spin-orbital coupling if it can help you to jugde my level on the subject. Thank you in advance for your answer.
It's explained in the first link you gave in the previous paragraph:

"The rotational quantum states of the asymmetric rotor, i. e.,
water, can be classified by $J_{K_{a}K_{c}}M$, with the total angular
momentum quantum number $J$, the projection labels $K_{a}$
and $K_{c}$ onto the molecule-fixed a and c axes as defined in
Figure 1 respectively, and the projection quantum number
$M$ onto the space-fixed $Z$ axis."

Do you know how angular momentum in quantum mechanics works? You can look at the projection of the angular momentum onto a specific axis and the total angular momentum. Turns out the Heisenberg Uncertainty Principle prevents you from knowing the angular momentum projections along more than one axis at once...