Structure of the periodic table

The periodic table is mostly determines by the electronic structure of atoms (see Atomic physics). See also Chemistry

There are three rules of thumb, which were discovered phenomenologically (I think), but are justifiable from quantum mechanics:

Aufbau principle: Shells should be filled starting with the lowest available energy state. An entire shell is filled before another shell is started.

Madelung’s Rule: The energy ordering is from lowest value of $n + l$ to the largest; and when two shells have the same value of $n + l$, fill the one with the smaller $n$ first.

(Mandelung's rule)

Hund's rules

Teaching Atomic Structure: Madelung’s and Hund’s Rules in One Chart

The Atomic orbitals are organized in Electron shells, and subshells. The set of occupied orbitals, corresponding to the electronic state of an atom (or molecule), is known as electronic configuration

Hydrogen