Fermions must be distinguishable in at least one quantum number.
Named after the austrian scientist Wolfgang Pauli (1900-1958) who invented it in 1925, and went Nobel in 1945.
What it means
First: Quantum mechanics is a curious little piece of science1 that states that every particle can be described by integer multiples of quantum properties like: Their spin, their mass, their movement energy, their vibrational energy, their rotational energy and some other more (which work like a grid, and are therefore called 'quantised'2.
To properly explain the Pauli principle a little quantum mechanics of spins should be known (or refreshed). Spin is an intrinsic property of a particle (like the rotation of a planet) but it has nothing to do with spinning - it was called that way for historical reasons. Like every quantum mechanical property, the spin is described by a function - obvious, and more precisely by a wave-function. But then again this is sort of irrelevant because the spin-function is NOT known (and it does not have to be known). The only thing necessary to know, is that this property exists as an integer multiple of a number that is for convenience taken to be 1/2 (The difference between two spins should be 1 and the 0 should be in the middle) Again: Just for convenience.
Smart scientists in the beginning of the 20th century went on to calculate systems with many particles in a quantum mechanical approach. Many experiments proved that electrons would at most show up in pairs (and not three or four). Looking at all the known quantum numbers of each electron in an atom, there is always at least one difference somewhere, the tiniest of them being the so called spin.
Stern and Gerlach (two very smart scientists of Munich, Germany) found out that the spin of one electron is always either 1/2 or -1/2 (and not 3 or 7/2 or 2). This sets the basics for the Pauli principle. Pauli found out that the reason for that lies in the (very complicated) mathematics of those systems3, and that this rule would not only be valid for electrons but for any particle with an odd multiple of the spin. Wild theory, but up to now it works perfectly fine.
According to Pauli's principle a different statistic would apply to particles for odd and even multiples of the spin:
Fermi-Dirac statistics for particles obeying Pauli's principle (i.e. odd multiples -- 1/2, 3/2, 5/2 etc...). Every state can only be populated by two particles.[continue here] The particles are therefore called fermions. Some important fermions are: quarks, prontons, neutrons and electrons.
Bose-Einstein statistics for the ones that must not obey Pauli's rule (i.e. even multiples 1, 2, 3 etc...) and can thus populate any state [continue here]. These particles are called bosons. Some prominent bosons are:4He-Nucleus, photons and gluons.
The particular application of Pauli's principle on electrons in atoms is also known as the 'Pauli Exclusion Principle' and it comes handy when calculating the electronic configuration of atoms and molecules:
Only two electrons can be in one orbital, the difference being their spin-quantum-number
To remind: The Pauli Principle is a wild piece of theory, but it was found to work for all fermions. That is, also for subatomic particles such as quarks. It is therefore a very fundamental principle, and not just a nice way to explain the electronic configuration of atoms.