What is particle spin in terms of waves? If particles are waves, then what really is spin?
 A: It is a misleading statement to say particles are waves and has created much misery in my opinion.
The physical system whose classical limit is a particle is called a "one-particle state" in quantum mechanics. In the hindsight, one can say that the classical description of a particle is given by listing the largest set of independent observable quantities associated with the particle. 
The description of the quantum state is given by describing probability amplitudes corresponding to different possible results of measurement of the largest set of quantities that are allowed to be measured simultaneously. 
So, for example, the description of a one-particle quantum state in reference to the measurement of its position is given by the list of probability amplitudes $\psi(x)$ where each complex number $\psi(x)$ is the probability amplitude of finding the particle to be at $x$ upon measurement. The shape of the probability function $|\psi(x)|^2$ is often like a "wave" when plotted against $x$. This is what supposedly motivates people to say that "particles are waves". It simply means that the probability of finding the particle's position varies along with positions in a wave-like manner. 
Now, recall my description of how a quantum state is described. In reference to that, the story about $\psi(x)$ is to be seen as the description of probability amplitudes corresponding to different results (the different values of $x$) of the measurement of position. But, the full description of a quantum state entails a description of such probability amplitudes corresponding to different results of all the measurements that can be simultaneously performed on a quantum state. The list of complex numbers $\psi(x)$ tells us the story of only one measurement, the position measurement. Could there be other measurements that can be performed along with the measurement of position? Turns out, yes. There is a measurement called the measurement of spin that one can perform on the quantum state while also performing the measurement of its position. Thus, rather than writing only the probability amplitudes $\psi(x)$ of finding the particle at a position $x$, we describe the quantum state via specifying the probability amplitudes $\psi(x,s)$ of finding the particle at a position $x$ with the value $s$ for the outcome of the spin measurement. 
I will not go into the details of what this measurement of spin is or what values $s$ are possible for the outcome of these measurements, etc. I intended to explain only why the description of spin in quantum mechanics is not at odds with anything else in quantum mechanics and why the spin isn't an ad-hoc addition to the framework of quantum mechanics. 
A: Waves can have spin. For example circularly polarized light has intrinsic angular momentum. 
A: I will try to clarify a few aspects that I consider being overlooked in some of the answers to the related questions (and links therein).
Particles are not waves
First of all we should make a distinction between what quantum aspect we are considering; the correct description of the universe
is quantum field theory, however for the sake of simplicity we may restrict ourselves to quantum mechanics, even if it is not 
relativistically covariant. 
In quantum mechanics particle dynamic is described by the Schrödinger equation: such equation is not Newton's-like and it is not a wave equation either; therefore particles are neither waves nor little balls: they are whatever it is described by the Schrödinger equation. However, one can show that experimentally the solution of such equation can describe (under some suitable boundary condition) the probability density to spatially find the particle here or there (see double-slit experiment or similar). Since the solution of the initial equation exists everywhere, in general, (or if not everywhere at least within the domain of definition of the specific operators) the probability density of spatially finding the particle here or there is non-local, which means there is a non-zero probability to find the particle anywhere in the universe no matter what dynamics you are describing. People misleadingly call this phenomenon "particle are waves" - but they are not, the interpretation being exactly what I mentioned above.
Orbital angular momentum
Start from the definition of angular momentum of a point-particle as $\mathbf{r}\times\mathbf{p}$: one can see that the quantum mechanical equivalent of such definition is so that the operators describing it must commute in a special way, in particular we have $[L_i, L_j] = \epsilon_{ijk}L_k$, where $\epsilon_{ijk}$ is the Levi-Civita symbol. In general a system is not only described by its position, rather also by its velocity, therefore a quantum system of a single particle (in some reference frame) is fully described by its spatial wave degree of freedom together with its angular momentum; for each position one could in principle have different values of the angular momentum: one refers to this characteristic as to "degeneracy" of a single quantum state.
Spin
Assume we know nothing about spin. We take a particle and we put it in an electromagnetic field just to see what happens, since we know how the particles are supposed to behave in there. The Stern and Gerlach experiment showed that, accidentally (nobody expected it), the energy levels arising from particles in a magnetic field could not be explained by only considering their position and angular momentum: there must have been something else contributing to the sub-division of said levels. People tried to put this "something else" manually into the equation and it turned out that if you assign an additional degree of freedom to the electron, with the constraint that such degree of freedom accidentally behaves like an angular momentum (see explanation before), all calculations work out perfectly. We do not know why it is so but we are forced to claim that particles possess an intrinsic quality which allegedly behaves as if it were a rotation, althoug it is not. People refer to it in the literature as to the "spin" of the particle.
