Pauli's Exclusion principle

Pauli exclusion principle states that two or more identical particles with half-integer spins (e.g. fermions) cannot simultaneously occupy the same quantum state within a system that obeys the laws of quantum mechanics. In the case of electrons in atoms, the exclusion principle can be stated as follows: in a poly-electron atom it is impossible for any two electrons to have the same two values of all four of their quantum numbers.

So, if hypothetically two fermions were in the same state, for example, in the same atom in the same orbital with the same spin, then interchanging them would change nothing and the total wavefunction would be unchanged. However, the only way a total wavefunction can both change sign (required for fermions), and also remain unchanged is that such a function must be zero everywhere, which means such a state cannot exist.
(Needs more explanation)