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` ).