The spin-statistics theorem is a theorem of quantum mechanics that relates the spin of a particle to the statistics obeyed by that particle. Spin is an intrinsic angular momentum of an elementary particle such as an electron. All particles have either integer spin or half-odd integer spin (these values are in units of Planck's constant divided by twice Pi). These two classes of particles are known respectively as fermions and bosons.
The theorem states that fermions are subject to the Pauli exclusion principle, while bosons are not. This means that only one fermion can occupy a given quantum state, while the number of bosons that can occupy a quantum state is not restricted. The basic building blocks of matter such as protons, neutrons, and electrons are fermions. Particles such as photons, which mediate forces between matter particles, are bosons.
There are a couple of interesting phenomena arising from the two types of statistics. The Bose-Einstein distribution which describes bosons leads to Bose-Einstein condensation. Below a certain temperature, most of the particles in a bosonic system will occupy the ground state (the state of lowest energy). Unusual properties such as superfluidity can result. The Fermi-Dirac distribution describing fermions also leads to interesting properties. Since only one fermion can occupy a given quantum state, the lowest single-particle energy level can contain two fermions, with the spins of the particles oppositely aligned. Thus, even at absolute zero, the system still has a significant amount of energy. As a result, a fermionic system will exert an outward pressure. Even at non-zero temperatures, such a pressure can exist. This pressure is responsible for keeping certain massive stars from collapsing due to gravity. For more on this, see white dwarf, neutron star, and black hole.