Fermions are the class of particles that make up matter, including electrons, quarks, and protons. They are defined by having half integer spin, and by obeying a strict rule called the Pauli exclusion principle, which says no two identical fermions can occupy the same quantum state at the same time.
That single rule has enormous consequences. It is the reason electrons stack into distinct shells around an atom’s nucleus rather than all collapsing into the lowest energy level, which is why the periodic table has the structure it does and why chemistry works the way it works.
Fermions stand in contrast to bosons, the other broad category of particles, which include force carriers like the photon and can pile into the same state without any restriction. That distinction between fermions and bosons is one of the deepest divides in particle physics.