Fermions include elementary particles, for example electrons or quarks, but also composite particles such as protons. In more vivid terms, Fermions are particles that make up matter, while, for instance, bosons are responsible for transmitting the elementary forces between particles and belong to a different class.
Fermions are subject to the Pauli exclusion principle. Two electrons can never occupy the same quantum state. This property is decisive for what we call matter: The fact that it is impossible for all electrons of an atom to occupy the lowest energy state, close to the atomic nucleus, but that instead the electrons have to spread out and occupy different states, leads to the differences between atoms. This in turn is the basis for the different chemical properties of elements.