Types of quantum field thories
A fermion is a particle/quantum field that obeys Fermi-Dirac statistics? (Pauli exclusion principle). By the spin-statistics theorem, this is the same thing as a particle whose spin is half an integer but not itself an integer.
In the standard model of particle physics and the standard model of cosmology fermions are the matter constituents of the observable universe, as opposed to bosons, which are the quanta of force gauge fields.
The formalization of fermions is in supergeometry: fermion fields are sections of spinor bundles with odd-degree fibers in supergeometry. This is exhibited by what is called Fermi-Dirac statistics? and the Pauli exclusion principle.
Fermions are named after Enrico Fermi.
For accounts in mathematical physics see for instance