"What is spin?" is a question of the species "What mass, charge, or color is?" and the answer is the same: "A fundamental property of elementary particles". I think that this property had been observed for the fist time in the famous "Stern–Gerlach experiment" or, according to wikipedia, "in the context of the emission spectrum of alkali metals".
The first attempt to theorize spin was done by W. Pauli, although, except than a two-valued degree of freedom, he had no idea what was it exactly. Some physicists thought that the fundamental particles were actually spinning, so they had some internal angular momentum, but this idea was abandoned at once, because the velocity required to have the observed value of angular momentum exceeds the speed of light.
The correct theoretical treatment of spin arises from the combination of QM and special relativity. Specifically, when you calculate the angular momentum of a particle in the rest frame of that particle, you find that it can have a non-zero value. That means that that the particle carries some angular momentum that is not due to its motion, i.e. spin is an internal amount of angular momentum.
Spin is quantized, that is it can have only certain values, and these values can be integer or half-integer multiples of \hbar. According to QFT and specifically because of the “causality principle”, particles of half-integer spin have to be “fermions” and those of integer spin have to be “bosons”, that is, a large number of identical particles have to obey “Fermi-Dirac” or “Bose-Einstein” statistics, respectively.
Those are some general information about spin and I hope will help you clear up the concept of spin.
