When one speaks of quantum particles he refers to particles that have dimensions on the atomic scale. At this scale, the usual physical properties, as we know in the real world, like energy, angular moment magnitude and its orientation as a vector, etc, comes in small and finite portions. Basically a quantum number is an integer that numbers these portions.
Originally, the quantum number was used to show how many portions of a finite and very small energy (`h/(2*pi)` ) a particle has. A smaller energy than the value of one portion can not exist. Usually this energy is absorbed and/or released by an electron from an external photon, hence its name of "quanta" (or as said, coming in portions). This was the primary type and was named the principal quantum number (`n`) .
Then there were postulated some other quantum numbers that numbered other physical quantities. For example the angular momentum of the electron was numbered, being shown that it also can take only integer values of the same quanta `h/(2*pi)` . This was the angular quantum number (`l` )
A third quantum number was attached to the projection of the angular momentum above, on to a certain axis (for example z axis). Thus the orientation of the above vector `L` could take only some numbered and fixed positions position in space. This was the magnetic quantum number (`m_l` )
`L_z = m_l*h/(2pi)`
Finally, a quantum number has been attached to the projection of spin on to a certain axis. Thus also the orientation of spin in space could take only some finite and fixed positions. This was named the spin quantum number (`m_s` ).
In Particle Physics and in chemistry electrons have a set of quantum numbers.So Quantum Numbers describe the energy of an electon and it's position in the atom.So basicacally knowing the fact that an electon has it's energy while spining around the nucleus of the atom,this energy however couldn't be dected till 1913,by using machines that detected the spectrum of hydrogen,it was discovered that this spectrum is the same as the spectrum lines from our sun.
But since we know that hydrogen gas has only electron and it's really easy to measure it's spectrum the question was can this be the same for different atoms that have more than one electron.By using other atoms with a higher number of electons they discovered that the spectrum lines changed,the model of sub-shell was born,now the main problem was to build a system that can tell how electons are arranged in an atom, this system was called the quantum system which consists of 4 set of numbers.
First Quatum Number is n Discribes the atom shell,or the energy level.It can obtain these values 0,1,2,3,4,5,6 It migh change in the future if a new period is added to the Perioic Table than there will come a number 7, so this is the basic quantum number.
Second Quantum Number is l Describes the number of Subshells,it can obtain these values 0 = s , 1 = p, 2=d , 3 = f.
Third Quatum Number is ml The values of mℓ range from −ℓ to ℓ. The s subshell (ℓ = 0) contains only one orbital, and therefore the mℓ of an electron in an s subshell will always be 0. The p subshell (ℓ = 1) contains three orbitals (in some systems, depicted as three "dumbbell-shaped" clouds), so the mℓ of an electron in a p subshell will be −1, 0, or 1. The d subshell (ℓ = 2) contains five orbitals, with mℓ values of −2, −1, 0, 1, and 2.
The fourth quantum numbers is ms The fourth, ms, describes the spin of the electron within that orbital.The spin is +1/2 and -1/2 thus it is imposible for an orbial to hold more than 2 electrons because they should have different spin numbers.