Spin is not a simple concept if you don't have the proper mathematical tools at hand. In this case, it's representation theory of the rotation group, SO(3), or in quantum theory its covering group, the SU(2). Quickly stated: The spin describes the behavior of an asymptotic-free single-particle state for such a particle at rest (vanishing momentum).
The best I can provide is to stick to phenomenology. The spin of charged elementary particles brings with it an intrinsic magnetic dipole moment. This is in a true sense an elementary magnetic field this particle (at rest!) intrinsically posesses as much like the electrostatic field it posseses due to its electric charge.
Further, also the electromagnetic (quantum) field is a dynamical entity in its own right. You cannot say the electric or the magnetic field are derived from each other but, according to the theory of relativity, these are six components of the electromagnetic field. Which of these components you call "electric" or "magnetic" field depends on the reference frame you choose, as much as which component of a vector you call the "x direction", depending on the (Cartesian) basis you choose to describe it.
The issue of spin becomes even more complicated when it comes to non-elementary particles, like the hadrons. A good example is, as already mentioned, the neutron, which has a vanishing net charge but consists of a very complicated state of quarks and gluons, which are bound by the strong force. The exact nature of this binding is not yet fully understood. In the naive parton picture you may say it consists of three quarks (one up and two down quarks), but these "valence quarks" are not the point particles you describe as Dirac fields in the QCD lagrangian. These socalled "current quarks" carry a mass of a few MeV, as inferred from lattice QCD and chiral perturbation theory compared to the explicit breaking of chiral symmetry (e.g., the finite value of the pion mass). This is a complicated issue in itself. A good review can be found in the particle data group's review of particle physics,
http://pdg.lbl.gov/2013/reviews/rpp2012-rev-quark-masses.pdf
Thus already the mass of the nucleon (proton or the neutron) is pretty complicated to understand. Only about 2% is due to the current-quark masses (i.e., the Higgs mechanism) the rest is due to the strong interaction in terms of a cloud of virtual gluons and "sea quarks" all together making up the nucleon.
The more complicated is the spin. It's not even easy to say, how to define the spin of the constituents and how the measured spin 1/2 of the nucleons is shared by them. This is ongoing research. The same holds for the associated magnetic moment of the nucleons, which after all are not elementary but complicated composite objects of finite extent as described above.