To understand the Pauli exclusion principle you just need to know how states are calculated in quantum mechanics.

A state is simply the way something is. Every unique arrangement of a system (say an atom) is considered to be an individual state. States are labeled by the quantum numbers associated with that arrangement of the system. Say you have a Hydrogen atom with the electron in it's lowest energy. You hit the electron with a photon, it gains energy, so the principle quantum number(the quantum number that describes the energy of the electron) increases. Before and after the photon hits are two different states whoose quantum numbers have different values.

Now something else you need to know about the quantum numbers is that they are arguments to a function. The function is called the Wavefunction of the system and it describes the system completely. Lets call the quantum numbers n,s,l,m then f(n,s,l,m) = state of the system where f is the wavefunction.

Often we want to know how likely a certain set of quantum numbers is to appear in a system, i.e. how likely the system is to be in a certain state f. If i have an ultraviolet photon and I chuck it at my hydrogen atom how likely am I to ionise the atom, for example.

Quantum mechanics describes a formalism for accurately calculating the probabilities that any state of a system exists. As input you feed in the wavefunction with the appropriate quantum numbers and as output you get a probability. The machine that converts the quantum numbers to a probability is an integral or more properly an overlap integral. I can't format integrals in HTML, but let's call the funny s shaped part I. The integration is over all of space so we will call the domain dx the integral looks like

I f(n,s,l,m)f*(n,s,l,m) dx .

The * indicates that we take the complex conjugate of the wavefunction when calculating the probability and the reason we do this is to ensure that the result will be positive definite+ (the function can sometimes be complex).

You don't really have to worry about that to understand the Pauli exclusion principle. The exclusion principle comes about because the wavefunction for an electron is always anti-symmetric. This is the definition of a Fermion. When you have many electrons and you you try to write down the probability of two of them being in exactly the same place then you have to calculate the overlap integral but the function f is strictly anti-symmetric. Now from basic calculus you should learn that the integral of an anti-symmetric wavefunction is 0!. The probability of two electrons being in the same sate is 0. It can never happen, it is not even that it is very unlikely, it is actually impossible. The magic of the Pauli exclusion principle comes about because of the anti-symmetry of the electron's wavefunction.

By contrast an object which has a symmetric wavefunction will quite like to be in the same state as another similar object. Particles with symmetric wavefunctions are called Bosons. Bose noticed this behavior and wrote a letter to Einstein about it. Einstein got the work published (Bose was pretty unknown at the time) and this work is the basis behind Bose-Einstein condensates.

+ This was stumbled upon in the 1926 by Max Born when people didn't really know what was going on

! The positive part of the function exactly cancels the negative part of the function when you calculate the area under the function.