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.