The numbers give you the basic ground state electron configuration of the atoms by telling you how many electrons are in each shell.
An electron's 'position' in an atom can be specified by four quantum numbers:
- Principal quantum number ($n$). This tells you which shell the electron is in and can take values $1,~2,~3,~4~...$. The maximum number of electrons in each shell is given by $2n^2$ so the first shell can hold 2 electrons, the second 8, the third 18, the fourth 32 etc. The numbers given in your periodic table are the total number of electrons in each shell in the ground state (lowest energy state) of that element.
- Azimuthal quantum number ($l$).
- Magnetic quantum number ($m_e$).
- Spin quantum number ($m_s$).
See this wikipedia article for more detail on the meaning of quantum numbers.
Electron configurations can be written in more detail using $s,p,d,f$ notation. See this question for more detail.
