What is meant by the spin of a particle? I have been studying that electrons have quantum number called spin quantum number(s), this number can have either +1/2 or -1/2 value. If s=+1/2, the spin is clockwise and if s=-1/2, the spin is anti clockwise about it's imaginary axis.
But, I am facing some problems now with this concept, the problems are, photon have 1-spin, another recently discovered sub-atomic particle having spin 3 (para. 7), how physicists explain these spin?
 A: Spin arises from the need to represent the rotation group $\mathrm{SO}(3)$ upon our Hilbert space of states. We need such a representation because the rotations (together with space translations) correspond to the non-relativistic changes of reference frames.
Since states are only determined up to rays in the Hilbert space, the true space of states on which we must represent the group is the projective Hilbert space, and the projective representations of a Lie group are (under some conditions) in bijection to the linear representations of its covering group, which is $\mathrm{SU}(2)$.
It turns out that these representations can be labeled by an integer $s \in \mathbb{N}$ or a half-integer $s \in \mathbb{N} + \frac{1}{2}$. This number is what we call spin.
A: Spin is best understood as an intrinsic angular momentum. It is probably easier to understand the concept for a charged particle. A classical charged particle moving along a circle has an angular momentum and the "circuit" has a magnetic moment. Further, the two are proportional to each other. 
It is experimentally found that a charged particle like an electron has an magnetic moment, the way it has a charge and a mass. We therefore suggest that the electron also has an intrinsic angular momentum $\vec{S}$, proportional to its magnetic moment $\vec{\mu}$.
We also find experimentally that an electron orbital angular momentum $\vec{L}$ is not a conserved quantity but $\vec{J} = \vec{L} + \vec{S}$ is. Therefore, $\vec{S}$ is not just a mathematical convenience but a "real" angular momentum.
