Pauli’s exclusion principle is very important. It states that no two fermions can have the same set of four quantum numbers. These four numbers are the principle (n), orbital angular momentum (l), magnetic (m) and spin quantum numbers (either s or ms). For the exercise it is easiest to forget they have real physical significance and think of them as coding for the electrons in the shells of an atom. The numbers can take the values:
n= 1,2,…→ ∞ l=0→(n-1) m=-l→0→l s=+½,-½
You select a value of n, representing what shell you are looking at, and then see how many combinations come as a result. E.g. n = 1 means that l = 0, m = 0 and s can always be +½ or -½. This means there are only two possible combinations: 1,0,0,½ and 1,0,0,-½. And in the first shell there can only be two electrons maximum corresponding to helium. Next pick n = 2. Now l = 0 or 1, m = -1, 0 or 1 and s remains the same. The combinations are as listed: 2,1,1,½; 2,1,0,½; 2,1,-1,½; and all of these again with spin -½, so six in total. But when l moves down to zero you get 2,0,0,½ and 2,0,0,-½. These are very similar to the first shell and represent the sub shell, making eight spaces in this shell and of course eight electrons can fill the second shell of an atom.
Now Pauli’s exclusion principle is often just associated with the electrons in a single atom, claiming that only the electrons in this certain atom have to have different energy values. This is actually not true, every atom bound electron in the universe has a different energy value, a slight difference, but always a difference. This means that when you put two hydrogen atoms together one energy level will sit above the other and scince all electrons want to be in the lowest energy state the upper one will fall down formin a pair in the lower orbital:
Image from http://photonicswiki.org
As you can see the lower level is called the bonding layer and the upper is the antibonding layer. Now this example is from when you have just two hydrogens comeing together. When you have aproximately a quadrillion (1024 long scale) copper atoms in a piece of wire there are so many energy levels so close together they become indistinguishable and so are known as bands. Many of the electrons fall into the lower half of the band which becomes known as the valence band and the conduction band, the upper half, is left relatively empty. Except in copper these bands are so close together as to be overlapping. At room temperture electrons are easily promoted into the conducting band makeing copper a conductor. Diamond, however, isn’t a conductor. This is beccasue the gap between its conducting and valence bands is massive and no electrons can become free charge carriers. Semiconductors often sit in the middle where only some electrons can be promoted in small numbers:
Image from HyperPhysics
The exact position of these bands is down to the structure and formula the molecules take when they bond, but from a physics stand point by knowing where these bands line up completely determines the electrical properties of all materials.