Imagine you had a Helium atom with only one electron. Its binding energy is 53.4 eV (i.e. You would need to add 53.4 eV to remove this electron from the n = 1 state). Now you add a second electron. Think about the attractive and repulsive forces that will act on the second electron and try to answer the following questions:
Can this second electron go to the lowest energy state? Will it also have the 53.4 eV binding energy? What happens to the energy? If we move onto the element Lithium we see that the quantum number and electron spin parameters dictate that n = 1 is filled with the first two electrons and the third electron must go in n = 2.
Beyond Lithium
n = 2, l = 0, m = 0, s = ± (2 elements)
n = 2, l = 1, m = -1, 0, 1, s = ± (6 elements)
This gets us to Neon.
So what is the connection between quantum mechanics and the periodic table?
The properties of all of the elements in a column are similar. Most reactive elements on the far left, non reactive on the far right. Properties are related to the outer most electrons which can interact easily with other atoms.
Go to the following web site, it has a periodic table that lists the orbitals, number of electrons in each shell. Simply move your cursor over the element you are interested in.
http://www.dreamwv.com/primer/page/s_pertab.html
Have a look at elements in the same column, do they have orbital similarities?
The periodic table was mostly developed before folks knew about electrons and atoms. Its arrangement was developed by Mendelev, an early chemist, based on similarities in chemical reactions and physical properties such as melting and freezing points. It is only since the development of quantum mechanics that we understand why the elements behave in the way they do.
Have a look at this web page. It includes an animation which traces out the shapes of the electron orbitals. Remember the orbitals are not actually a path followed by the electron but a probability density. The animation on this web page is slightly deceiving.
http://www.colorado.edu/physics/2000/applets/a2.html