quantum numbers
If every orbital in an atom needed its own street address, the quantum numbers are that address — a short set of whole (or half) numbers that completely pins down which orbital an electron is in and which way it spins. Just as a postal address narrows from country to city to street to house, the quantum numbers narrow from shell to subshell to specific orbital to spin.
There are four. The principal quantum number n (1, 2, 3, ...) sets the shell and the main energy and size: bigger n means a bigger, higher-energy orbital further from the nucleus. The angular-momentum (azimuthal) quantum number l (0 up to n-1) sets the shape and the subshell: l = 0 is an s orbital, l = 1 is p, l = 2 is d, l = 3 is f. The magnetic quantum number m_l (from -l to +l) picks the orientation in space — for p (l = 1) there are three values, giving the three p orbitals. The spin quantum number m_s (+1/2 or -1/2) describes the electron's two-way intrinsic spin. No two electrons in one atom may share all four — that is the Pauli exclusion principle, which is exactly why each orbital holds at most two electrons (same n, l, m_l, but opposite spin).
These four numbers are the bookkeeping behind the whole periodic table: the allowed combinations explain why the first shell holds 2 electrons, the second 8, why there are exactly 5 d orbitals (giving the 10-wide d-block of transition metals) and 7 f orbitals (the 14-wide f-block). They are not free choices — they fall out of solving the quantum-mechanical equation for the atom, which is why they come in such rigid, integer steps.
An electron in a 3d orbital has n = 3, l = 2; m_l can be any of -2, -1, 0, +1, +2 (five orbitals); and m_s is +1/2 or -1/2.
Four numbers = one electron's complete address.
Electron "spin" is an intrinsic quantum property with no real spinning-top inside — the name is a historical analogy. It is genuinely two-valued, and that two-ness is what makes orbitals hold pairs.