In a nutshell, particles "are" unitary irreducible representations of the Poincaré group, which is the isometry group of Minkowski spacetime on which it acts transitively. Such representations can be constructed using the method of induced representations (cf. Wigner, Bargmann, Mackey,...) as classical fields on Minkowski spacetime subject to certain field equations: wave, Klein-Gordon, Weyl, Dirac, Maxwell, Rarira-Schwinger,... Mathematically these are sections of homogeneous vector bundles associated to certain finite-dimensional representations of the "little group", which is (the maximal compact subgroup of) the stabilizer (in the spin cover of the Lorentz group) of a point on the "mass shell" (=the momenta $p$ with $p^2 = - m^2$, where $m$ is the mass of the particle). The little group for massive representations is isomorphic to $SU(2)$, whereas that of massless particles to a nontrivial double cover of $SO(2)$. Hence massless particles are defined by their helicity (the label of the $SO(2)$ representation from which one induces) and massive particles by their mass and their spin (the label of the irrep of $SU(2)$ from which one induces). The covariant field equations are (the Fourier transform of) the projectors onto irreducible components.

Let's discuss the massive case, since your question mentions spin explicitly.

There are two kinds of irreps of $SU(2)$, those where $-1$ acts trivially and those where it does not. The former case are the integer spin representations whilst the latter are the half-integer representations.

The integer spin representations of $SU(2)$ are contained in the tensorial representations of the Lorentz group, whence this is why the fields for integer-spin massive particles are tensorial.

The irreps of $SU(2)$ with half-integer spin, those where $-1$ does not act trivially, are not contained in tensorial representations of the Lorentz group, since on these representations $SU(2)$ acts by conjugation and $-1$ acts trivially. In order to describe such reps in terms of fields you need to consider spinor fields, which are sections of spinor bundles (possibly twisted by tensor bundles for higher spin fields).

You can read about spinor bundles on any book in Spin Geometry. For example, there's a book by Lawson and Michelsohn of that name. Green-Schwarz-Witten's string theory book (second volume) has a physicsy discussion of this. They will also define the spin connection, which is a connection induced by the Levi-Civita connection on any "spin bundle", which is a principal fibre bundle lifting the bundle of oriented orthonormal frames in such a way that the bundle map between them restricts to the spin covering $\mathrm{Spin} \to \mathrm{SO}$ on the fibres.

It is not hard to show that the connection one-form for the spin connection, when pulled back to the manifold by the lift of a local frame (hence a local 1-form with values in the orthogonal Lie algebra) agrees with the similar expression for the Levi-Civita connection, which is why many books perhaps do not go through the trouble of defining it properly. It also requires introducing quite a bit of formalism, to which many physicists are allergic; although increasingly less so.