Quantum numbers arise when we try to find the solution of the Schr'dinger equation. Depending on the complexity the situation, determines how many quantum numbers there will be.
For example for a harmonic oscillator (a 1-D particle pond by a potential V=1/2kx^2 ) we only get one quantum number n and it represents the energy level of the harmonic oscillator. E=nhf.
For the 3-D one electron atom we get three n,l and ml. These represent the energy level, angular momentum and the angular moment in the z direction.
Another quantum number is ms and it represents the spin of the electron.
So as you can see as we extend the complexity of the situation, we get quantum numbers for everything, which is measurable.
Each electron has a set of four numbers, called quantum numbers, that specify it completely; no two electrons in the same atom can have the same four. That's a more precise statement of the Pauli exclusion principle.
Primary quantum number, which is given the symbol n,n tells you which of the "main" energy levels you're in.
The second quantum number is known as l. A value of l=0 corresponds to s, l=1 is p, l=2 is d, and so forth.
The third is magnetic quantum number m.
l, along with n and the third quantum number, m, is responsible for determining the shape of an electron's probability cloud.
The fourth quantum number, s, does indeed pertain to an electron's spin.
s only has two possible values+1/2 or-1/2.s= +1/2 means "spin up" and s= -1/2 means "spin down."
Within the level given by a particular n, l can take on only integer values from 0 to n-1.So when n is 1, l can only be 0, and that's why the first row has only s states. Then when n=2, l can be either 0 or 1, and that gives you s and p. Given a particular l, m is entitled to be any integer from minus l up to l. For example, when l=1, m can be -1, 0, or 1; those are your three p states. For a given l, there are 2l+1 different values of m.
Yash Kumar, B.S., Physics graduate student, India