Quantum numbers are numbers involved in quantum mechanics to describe the state of physical systems.
In quantum mechanics, a large number of physical quantities are discrete rather than continuous. This is the case, for example, for energy and orbital angular momentum or even spin.
In a Bohr atom the energy levels associated with the electrons form a discrete series related to integers. Similarly, the angular momentum of this type of atom cannot vary continuously in orientation and magnitude in space, and here too integers are involved.
Spectroscopic studies have shown that there is a thin structure in the spectrum of the hydrogen atom, with energy sub-levels within the principle energy levels. And here again these levels are described by formulae involving a discrete series of half-integer numbers.
These numbers were baptised quantum numbers and the terminology is still used for other numbers forming discrete sets to describe the quantum states of physical systems. This is true for baryon, lepton and isospin numbers and for strangeness, for example, in the world of elementary particles.
The number n in Balmer's formula
for the Bohr atom is called the principle quantum number and the half-integer numbers for the fine energy levels are numbers related to the angular momentum of the electron, the spin.