Suppose a physical quantity like energy, angular momentum, spin etc., can have only some discrete values in a system, then those are called quantum numbers. Energy level of electrons in an atom of an element like silicon are supposed to be having a few values or ranges of values : 2 or 3 or 4 or 5 or 6 or 7 ... bands or so. Electron energy of the electrons in this material or other materials depends on which band or level it stays. There is an equation of quantum mechanics called shrodinger's equation that describes the properties of the particle. The finite number of solutions for this equation give the quantum numbers.
For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6.