Spin is indeed a pretty abstract subject, but it's fully understood within both nonrelativistic quantum mechanics and relativistic quantum field theory. In both cases it arises from the fact that for an inertial observer at any instant of time space is Euclidean and thus obeys rotational symmetry (around any fixed point). Thus a quantum theoretical description of a single particle must allow to define rotations, and the fundamental laws of nature do not depend on the orientation of the inertial reference frame.
Now you can analyse the representations of the rotation symmetry group within quantum theory. This is a pretty mathematical and abstract subject but at the same time belongs to the most beautiful ideas in theoretical physics ever. The analysis first tells you that not the classical rotation group is the fundamental way to describe rotations in quantum theory but its covering group, the SU(2). Then the mathematics of SU(2) tells you how the irreducible unitary representations on Hilbert space look like, and the spin defines to which representation the subspace of Hilbert space of a single particle with vanishing momentum belongs (I assume massive particles here, for massless particles it's a bit more complicated). Thus it tells you how the state of a single particle at rest transforms under rotations.
That's very abstract, as I said, and it's also a bit incomplete since a concept is only physically well-defined, if we can say how to observe and even measure it somehow. Fortunately, as is well known already from classical physics, angular momentum of charged systems implies the existence of a magnetic moment, and this magnetic moment can be measured by bringing the system (or in our case a single quantum particle) into an inhomogeneous magnetic field. One of the fundamental experiments done in the history of quantum physics is the Stern-Gerlach experiment, which explcitly shows the existence of the magnetic moment of the electron (more precisely of silver atoms, but it's total spin is due to the one valence electron in the atom's outermost shell) explictly, and it turns out to be indeed a spin 1/2 particle. This is something that is not in any way observable within classical physics and thus is, as stressed above, very hard to visualize intuitively. The idea of a point-like magnetic dipole, however, is, and on an operational level one can invisage spin as such.
In addition, the spin occurs in the angular-momentum bilance of reactions, and this shows that it is indeed a kind of angular momentum in the literal sense. So, spin is very well understood in both a very abstract level and well established by observation.
As bhobba said, the best book to learn about the non-relativistic foundation of quantum theory in symmetry principles is Ballentine's book Quantum Mechanics. For the same subject in relativistic QT, see Weinberg, The Quantum Theory of Fields 1.
