quantum number
Think of how a postal address pins down a single home: country, city, street, house number. Each part narrows things down until just one place remains. An electron in an atom has its own kind of address, made of a small set of whole or half numbers. Read them off and you know which orbital the electron is in, how big and what shape it is, which way it points, and which way it spins. Those numbers are the quantum numbers.
More precisely, quantum numbers are the integers (and half-integers) that label the allowed quantum states of a system. For an electron in an atom there are four: the principal quantum number sets the main energy level and overall size; the angular momentum quantum number sets the orbital's shape (s, p, d, f); the magnetic quantum number sets its orientation in space; and the spin quantum number gives the electron's two-valued spin. They arise naturally as the labels on the solutions of the Schrödinger equation.
The honest point is that quantum numbers are not arbitrary tags we attach for bookkeeping — they are forced on us by the maths, because only certain whole-number patterns let a wavefunction fit together properly. Their real power for chemistry comes from a companion rule, the Pauli exclusion principle: no two electrons in an atom may share the identical set of all four. That single restriction is what builds the entire structure of the periodic table.
The two electrons in a helium atom share the address n=1, l=0, m=0 — both in the 1s orbital. The Pauli rule then forbids them from also matching on spin, so one must spin up and the other spin down. With both spin slots filled, the 1s orbital is full, which is exactly why helium is a stable, unreactive gas.
Four numbers per electron, no two alike — the rule that fills the periodic table.
Spin is the odd one out: it has no everyday analogue and the electron is not literally a spinning ball. Its quantum number can only be +½ or −½, which is why each orbital holds at most two electrons, one of each spin.