In particle physics, a fermion is a type of particle that obeys the rules of FermiDirac statistics, namely the Pauli Exclusion Principle. These fermions also have a quantum spin with contains a halfinteger value, such as 1/2, 1/2, 3/2, and so on. (By comparison, there are other types of particles, called bosons, that have a integer spin, such as 0, 1, 1, 2, 2, etc.)
What Makes Fermions So Special
Fermions are sometimes called matter particles, because they are the particles that make up most of what we think of as physical matter in our world, including protons, neutrons, and electrons.
Fermions were first predicted in 1925 by the physicist Wolfgang Pauli, who was trying to figure out how to explain the atomic structure proposed in 1922 by Niels Bohr. Bohr had used experimental evidence to build an atomic model which contained electron shells, creating stable orbits for electrons to move around the atomic nucleus. Though this matched well with the evidence, there was no particular reason why this structure would be stable and that's the explanation that Pauli was trying to reach. He realized that if you assigned quantum numbers (later named quantum spin) to these electrons, then there seemed to be some sort of principle which meant that no two of the electrons could be in exactly the same state. This rule became known as the Pauli Exclusion Principle (see the About.com Chemistry definition).
In 1926, Enrico Fermi and Paul Dirac independently tried to understand other aspects of seeminglycontradictory electron behavior and, in doing so, established a more complete statistical way of dealing with electrons. Though Fermi developed the system first, they were close enough and both did enough work that posterity has dubbed their statistical method FermiDirac statistics, though the particles themselves were named after Fermi himself.
The fact that fermions cannot all collapse into the same state  again, that's the ultimate meaning of the Pauli Exclusion Principle  is very important. The fermions within the sun (and all other stars) are collapsing together under the intense force of gravity, but they cannot fully collapse because of the Pauli Exclusion Principle. As a result, there is a pressure generated that pushes against the gravitational collapse of the star's matter. It is this pressure which generates the solar heat that fuels not only our planet, but so much of the energy in the rest of our universe ... including the very formation of heavy elements, as described by stellar nucleosynthesis.
Fundamental Fermions
There are a total of 12 fundamental fermions  fermions that aren't made up of smaller particles  that have been experimentally identified. They fall in two categories:

Quarks  Quarks are the particles that make up hadrons, such as protons and neutrons. There are 6 distinct types of quarks:

Leptons  There are 6 types of leptons:
In addition to these particles, the theory of supersymmetry predicts that every boson would have a sofarundetected fermionic counterpart. Since there are 4 to 6 fundamental bosons, this would suggest that  if supersymmetry is true  there are another 4 to 6 fundamental fermions that have not yet been detected, presumably because they are highly unstable and have decayed into other forms.
Composite Fermions
Beyond the fundamental fermions, another class of fermions can be created by combining fermions together (possibly along with bosons) to get a resulting particle with a halfinteger spin. The quantum spins add up, so some basic mathematics shows that any particle which contains an odd number of fermions is going to end up with a halfinteger spin and, therefore, will be a fermion itself. Some examples include:
 Baryons  These are particles, like protons and neutrons, that are composed of three quarks joined together. Since each quark has a halfinteger spin, the resulting baryon will always have a halfinteger spin, no matter which three types of quark join together to form it.
 Helium3  Contains 2 protons and 1 neutron in the nucleus, along with 2 electrons circling it. Since there are an odd number of fermions, the resulting spin is a halfinteger value. This means that helium3 is a fermion as well.