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In quantum field theory on Minkowski space all fields must transform according to a definite (finite dimensional) representation of the universal cover of the Poincare group, which determines the spin of each field. From the representation theory of the Poincare group it is known that the spin $s$ is a number $s = \frac{n}{2}$ with $n \in \mathbb{N}$.
On the other hand, if we take fields to be pointwise localized in the sense of the Wightman axioms, then the locality axiom (also known as Einstein microcausality ) says that spacelike separated field operators either commute or anticommute: Two Fermionic fields anticommute, two Bosonic fields commute, a Fermionic and a Bosonic field commute.
The spin-statistics theorem states that fields with integer spin $s$ (n is even) are Bosonic fields, fields with half-integer spin (n is uneven) are Fermionic fields. A better name for the theorem would therefore be spin- commutation theorem, the name spin- statistics theorem stems from the fact that Bosons (the particles associated to Bosonic fields) are social, multiple particles can exist in the same quantum state, while Fermions are not social: The Pauli exclusion principle says maximally one Fermion can exist in a given quantum state. This leads to different partition functions in statistical mechanics of systems consisting of Bosons only and of Fermions only, hence the name.
The statement and proof of the theorem depend on the framework for quantum field theory that is used, therefore there are, strictly speaking, several versions of the spin-statistics theorem, but the physical interpretation is always the same.
In the Haag-Kastler approach the Bisognano-Wichmann theorem states a relation of the representation of the Poincare group on certain local algebras of local nets and their modular groups. If this relation holds for a given net, then this net is said to fulfill the Bisognano-Wichmann property.
The Guido-Longo-paper cited in the references states the spin-statistics theorem for local nets with this property.
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The original formulation of the spin-statistics theorem is due to
Markus Fierz, Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin. Helvetica Physica Acta. 12 (1): 3–37. (1939) doi:10.5169/seals-110930
Wolfgang Pauli, The connection between spin and statistics, Phys. Rev. 58, 716–722 (1940)
The classical textbook reference is
The proof, which goes back originally to Fermi. There is also a more intuitive approach based on topology. One can see hints of it in Feynman’s lecture here:
An exposition is at
A statement and proof of both a spin-statistics and a PCT theorem in the Haag-Kastler approach can be found in this paper:
Generalization to curved spacetime is in
Discussion relating the spin-statistics theorem to extended topological field theory, categorification (via 2-rings) and Deligne's theorem on tensor categories is in