I think, here is some confusion about "spin" in this discussion. First of all the spin of a particle, s \in \{0,1/2,1,3/2,\ldots \} defines the representation of the rotation group, which is a subgroup of both the Galilei symmetry of Newtonian space-time and Minkowski symmetry of special relativistic (Einsteinian) space-time.
Let's discuss the relativistic case right away since this was asked in the OP and let's concentrate on the massive case.
More specifically for massive particles, the spin is defined as the representation of the rotation group for a particle at rest in a given inertial frame of reference. This is the socalled "little group" of the massive representations of the unitary representations of the covering group of the special orthochronous Poincare group. Covering group means that in the semidirect product of translations and \mathrm{SO}(1,3)^{\uparrow} the special orthochronous Lorentz group is substituted by its universal covering group, which is \mathrm{SL}(2,\mathbb{C}). The representation of the little group induces the representation of the full group via the Frobenius construction (see Wigner 1939).
Now there are a special type of representations of the Poincare group in quantum field theory, namely those with local field operators, i.e., where you have field operators that transform unitarily under the Poincare group such as the analogous classical fields. This concept of locality is very important, because it's sufficient to construct microcausal theories, which lead to a unitary Poincare covariant S matrix, which defines the observables of quantum field theory like decay rates and cross sections (I omit the idea of observables within relativistic many-body theory, which is a quite straight-forward extension of the observables defined via the S matrix in vacuum QFT). Together with the boundedness of the Hamiltonian these concepts lead to the very successful QFTs describing the Standard Model of Elementary Particles and effective relativistic QFTs (e.g., to describe low-energy QCD in terms of (resummed) chiral perturbation theory for hadrons).
This very abstract mathematical considerations justify a shortcut, which is very valuable to introduce relativistic QFT to beginners in the field and coming quickly to the physical applications: canonical field quantization. This is a heuristical method to get to the QFTs underlying the standard model more quickly. The idea is to look for local Poincare invariant action functionals. Local means the action functional is derived from a Lagrangian that is a polynomial of fields and its first space-time derivatives such that the action functional is Poincare invariant. The classical field equations are then Poincare co-variant, where the Poincare symmetry is realized as (not necessarily in any sense unitary) local representations. E.g., for a vector field, the transformation under Boosts and rotations is given by
A'{}^{\mu}(x')={\Lambda^{\mu}}_{\nu} A^{\nu}(\Lambda^{-1} x'), \quad \Lambda \in \mathrm{SO}(1,3)^{\uparrow}.
Unter translations the fields transform as scalars.
To find all possible classical fields, including those where the \mathrm{SO}(1,3)^{\uparrow} is substituted by its covering group \mathrm{SL}(2, \mathbb{C}), it is sufficient to investigate the finite-dimensional linear transformations of this group (there are no unitary ones except the trivial one since \mathrm{SL}(2,\mathbb{C}) is not compact) or its Lie algebra. This Lie algebra is isomorphic to a direct sum \mathrm{su}(2) \oplus \mathrm{su}(2), but it's complexified. The rotations, as a subgroup of the Lorentz group can be represented unitarily with finite-dimensional representations, because it's a compact Lie algebra (all irreducible representations (irreps) are isomorphic to a unitary one). Due to the structure as a direct sum of two su(2) algebras, each irrep is characterized by two integer or half-integer numbers (s_1,s_2), each giving the representation for the two Lie algebras. For the rotation group as a subgroup, the representations are in general not irreducible, and to define elementary particles of definite spin, one has to project out all irreducible parts except the one you want to use to describe an elementary particle. The irreps. of the SU(2) as a subgroup are given by the rules of angular-momentum addition ("Clebsch-Gordan gymnastics"), i.e., it contains spin representations for s \in \{|s_1-s_2|,|s_1-s_2|+1,\ldots,s_1+s_2 \}. Projecting out the unwanted spin states for an elementary particle of definite spine is usually done with the field equations and appropriate constraints.
E.g., for a vector field you have four components A^{\mu}. A possible relativistically covariant equation is the Proca equation,
\partial_{\mu} F^{\mu \nu}=-m^2 A^{\nu}, \quad F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}
The vector field belongs to the (1/2,1/2) representation of the Lorentzgroup, i.e., it contains a s=0 (scalar) piece and an s=1 (vector) piece for the representation of the rotation group as a subgroup. To project out the unwanted scalar piece, the field equation is already sufficient (for the massive case only!), because the Proca equation implies the constraint \partial_{\mu} A^{\mu}=0. You are left with three field-degrees of freedom which represent the rotations in the s=1 representation 2s+1=3 is the dimension of this representation).
Now let's come to fields representing (in the quantized version) spin 1/2. The two most simple possibilities are the representations (1/2,0) and (0,1/2), which are both 2-dimensional and only contain the s=1/2 representation for the rotation subgroup. The associated \mathbb{C}^2 valued fields are called Weyl-spinor fields. In the standard model of (massless Dirac) neutrinos the neutrinos are represented by such Weyl fields.
Now to the Dirac fields: They are introduced to admit an extension of the representation of the proper orthochronous Lorentz group with spatial reflections (parity). One can prove that this can be achieved (uniquely) by the direct sum (1/2,0) \oplus (0,1/2). This makes a representation with two Weyl spinors, where the space-reflection trafo involves and interchange of these two Weyl spinors (the direct realization of this theorem is known as the Weyl or chiral representation of the Dirac algebra). That's why quarks and the massive leptons are represented by Dirac fields, because it admits parity conservation (space-reflection symmetry) for the strong and the electromagnetic part of the Standard Model. Only the weak interaction with its "vector-minus-axial vector" structure violates parity and thus is realized as a chiral gauge theory. Of course this simple picture is a bit complicated by the fact that in the standard model we need a realization as a Higgsed gauge symmetry leading to quantum flavor dynamics (Glashow-Salam-Weinberg model), which "mixes" (or in a certain weak sense unifies) the electromagnetic and the weak interactions.
For more on spinors and spinor fields, see my QFT manuscript
http://fias.uni-frankfurt.de/~hees/publ/lect.pdf
or Weinberg, Quantum Theory of Fields I;
Sexl, Urbandtke, Relativity, Groups, Particles.
