Decided to throw my two cents worth into the ring...
I see two aspects to the OP's question the first of which has been pretty well answered, i.e. spin is intrinsic angular momentum. For a classical analogue one need simply consider the Earth and Sun. The Earth has orbital angular momentum due to its orbit around the Sun but in addition has spin due to its rotation about its own axis.
The second aspect is the implied question from the quantum case "what is a spinor". I would begin with the canonical duality between observables and generators of kinematic transformations.
Spatial translations correspond to and are canonically generated by linear momenta.
Rotations correspond to and are canonically generated by angular momenta.
The intrinsic spin of a quantum particle then manifests as its representation under rotations. A spin 1 particle transforms as a vector, a spin 2 particle transforms as a symmetric 2-tensor and so on... then there are the spinor representations...
For all my study I didn't fully understand spinors until I studied Clifford algebras in detail.
Begin with the antisymmetric tensor representations of rotations (or Lorentz transformations in relativistic theory... or SO(p,n) in abstract generalizations).
For rotations in n dimensions you have scalar (dim 1=n choose 0) vector (dim n = n choose 1) bi-vector=rank 2 antisymmetric tensor (dim n choose 2) and so on up to rank n antisymmetric tensors (dim n choose n = 1).
In three dimensions these correspond to scalar, vector, bi-vector and pseudo-scalar of dimension 1, 3, 3, and 1 respectively. Now we can represent rotations of each of these by writing the corresponding rotation matrix acting on the corresponding column vectors but with Clifford algebras we have a nice
adjoint representation.
For n dimensional rotations we construct the Clifford algebra from (the identity 1 and) the grade 1 elements corresponding to a basis of the vector representation. \gamma_1, \gamma_2, \cdots \gamma_n
These have the property that their anticommutator is a multiple of the identity, and that multiplier is the metric:
\gamma_i \gamma_j + \gamma_j \gamma_i = g_{ij} \mathbf{1}
Then resolving all products of gammas into commutator and anti-commutator components we find that we have higher grade products corresponding exactly to higher rank anti-symmetric tensors. In particular the grade two elements \gamma_{ij} = \frac{1}{2}[\gamma_i, \gamma_j] are the generators of rotations in the i,j planes (or pseudo-rotations if the metric is indefinite).
One observes then that rotations acting on the clifford algebra are represented by the adjoint action, that is to say a rotation of any element A through angle \theta in the i,j plane is given by:
R_\theta[A] = e^{\theta\gamma_{ij}/2}A e^{-\theta\gamma_{ij}/2}
--The grade 1 elements \gamma_i will transform as if rotated as vectors.
--The grade 2 elements \gamma_{jk} will transform as bi-vectors.
...
--The grade k elements will transform as rank k anti-symmetric tensors.
and so on.
So the Clifford algebra represents rotations of vectors and tensors (including the rotation generators themselves) internally via this adjoint action. This is why Clifford algebras are also called
geometric algebras. Now this by itself is just a simple mathematical trick, like using quaternions or complex numbers to represent rotations (which is actually doing the same thing as these are clifford algebras when graded appropriately).
The real insight is when we express the elements of the Clifford algebra as matrices and then ask what is the vector space these matrices act on via left multiplication. The answer is that they are spinors.
R_\theta[\psi] = e^{\theta \gamma_{ij}}\psi
In this sense the space of spinors is the "square root" of the space of anti-symmetric tensors (including vectors) in the way that abstract vectors are "square roots" of matrices.
Cliff = \Psi\otimes\Psi^*
When all the mathematical construction is said and done, we have another set of irreducible representations of groups of rotations in the spinor space \Psi
For rotations in 3 dimension the Clifford algebra is the algebra generated by the Pauli spin matrices. The \sigma_i are the grade 1, vector elements. Their commutators [\sigma_i,\sigma_j]= 2i\sigma_k = 2\sigma_{ij} are the grade 2 generators or rotations. They are expressed as 2x2 complex matrices and the (right ideal) complex 2-vectors they act upon are the spinors of rotation in 3 dimensions.
For relativistic rotations and boosts we have the Lorentz group, the Clifford algebra of Dirac gamma matrices: \{1,\gamma_\mu, \gamma_{\mu\nu}, \gamma_{\mu\nu\lambda},\gamma^5=\gamma_{1234}\}
representable as 4x4 matrices acting on 4 dimensional relativistic spinors.
Getting back to what they mean physically, a point particle would have no intrinsic spin and is unaffected by rotation about its center of mass. Since this description relies on treating it as a classical particle we rather say in quantum mechanics, rotating a spin-0 particle about any origin is represented only by acting on the coordinates (x,y,z) of its wave-function.
With this then the observable of angular momentum is expressed wholly by the orbital component: L_z = -i(x\partial_y - y\partial_x) etc.
For a particle with intrinsic spin, one has a vector, tensor, or
spinor valued wave-function and the total angular momentum observable will be the orbital component plus the representation of intrinsic spin expressed by a matrix. In the spinor case that matrix will be S_{3}=-i\gamma_{12}=\sigma_3 et al.
The physics is then in the spectrum of the total angular momentum and its orbital and intrinsic components: J = S+L.
[Edit: I would recommend
https://www.amazon.com/dp/0521551773/?tag=pfamazon01-20 by Ian Porteous as a thorough advanced reference. ]
