Fermions are particles which have half-integer spin and therefore are constrained by the Pauli exclusion principle. Particles with integer spin are called bosons. Fermions include electrons, protons, neutrons. The wavefunction which describes a collection of fermions must be antisymmetric with respect to the exchange of identical particles, while the wavefunction for a collection of bosons is symmetric.
The fact that electrons are fermions is foundational to the buildup of the periodic table of the elements since there can be only one electron for each state in an atom (only one electron for each possible set of quantum numbers). The fermion nature of electrons also governs the behavior of electrons in a metal where at low temperatures all the low energy states are filled up to a level called the Fermi energy. This filling of states is described by Fermi-Dirac statistics.
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