Electrons as tiny magnets I recently was watching videos about magnets. A lot of them said that the movement of electrons around an atom would create a magnetic field which will be tiny. They also said that because the electron has spin that also adds to the magnetic field. These electrons have an intrinsic property called spin. They are also known as tiny magnets. 
What I don’t understand is how can a fundamental particle( say electrons) spinning around its axis generate a magnetic field?
And why are they known as tiny magnets( if so then can ‘North Pole and South Pole of this magnet’ be defined?
If special relativity is applicable, then how can it explain this phenomenon because it explains magnetic force as a result of length contraction between 2 moving wires. 

Correct me if I’m wrong. 
 A: The problem lies in the intuitive picture they gave you for spin, which was useful historically but is not accurate. 
Spin is an intrinsic property of particles, it does not mean that such particles are rotating on an axis. It was associated to this idea because mathematically, spin follows the same addition rules as angular momentum (which IS related to rotations). More precisely spin is described by irreducible representations of the group $SU(2)$. These exist for certain labels $j=1,2,3,...$ and the number $S^2=j(j+1)\hslash$ is what is usually referred as the spin of the particle. 
Now, how does this relate to the magnetic properties? Well one could say that given that it follows the same mathematical rules as angular momentum, an effect that is related to rotations might also be affected by spin. This behaviour was observed by the famous Stern-Gerlach experiment in 1922, which proves the existence of some intrinsic property of electrons which interacts with an electromagnetic field. They also found that this property, for the case of electrons, could only result in two values $\pm \frac{1}{2}\hslash$ leading to the picture you have of a magnet perhaps. However we actually treat particles to be point-like so (in principle) no size and therefore no labeling of North/South poles is possible. 
Curisouly enough, it is exactly special relativity which theoretically tells us about this property. Once physicist (namely Dirac) discovered the correct special relativistic equation followed by fermions (certain quantum particles such as the electron), the Dirac equation, they realized spin was already there. Specifically, Dirac equation comes as a vector differential equation, having as a solution a "wave function" with four components of which one can identify particle and antiparticle and their corresponding spin projections.
A: Magnets are macroscopic objects described by  classical electricity and magnetism , Maxwell's equations. 
Electrons are elementary particles posited in the standard model of particle physics axiomatically,  i.e. their properties have been measured experimentally and are accepted as input to the theory. They are quantum mechanical entities.
All classical theories emerge from the underlying framework of quantum mechanics.

What I don’t understand is how can a fundamental particle( say electrons) spinning around its axis generate a magnetic field?

The classical language is used, but the particles in the standard model table are point particles, their spin and charge have been measured by painstaking experiments, using quantum mechanics and conservation laws that the QM solutions have to obey. Conservation of angular momentum is the method of deriving the spin of elementary particles. The word "spin" does not imply spinning as for a macroscopic object. It is there as an identifier of conservation of angular momentum at the quantum mechanical level. (It is the same as the orbitals of the electrons, not orbits, which are probability loci for the electrons in an energy level of an atom.) The resultant magnetic moment is not the result of orbits, but of the probability distribution of the orbitals of the electron.

And why are they known as tiny magnets( if so then can ‘North Pole and South Pole of this magnet’ be defined? 

It is an analogy to the macroscopic magnets. Yes, north and south may be defined because they have two states of spin, +1/2 -1/2 . One has to keep in mind the quantum versus classical distinction though.

If special relativity is applicable, then how can it explain this phenomenon because it explains magnetic force as a result of length contraction between 2 moving wires. 

Special relativity is paramount in the quantum framework, and the corresponding equations for  generating magnetic fields exist, using the defined probability loci of the solutions for the equations.
Atoms and molecules and magnetic domains smaller than nanometers are modeled in the quantum mechanical frame. Long range behavior is built up by these underlying quantum states.
A: What you have heard are the first perceptions about the magnetic dipole of electrons. The explanation of the electrons electric charge was given by the separation of charges and Thomson's famous experiment.
Such experiment is not carried out for the magnetic dipole of separated charges. The usual experiment is the acceleration of charges which is accompanied by the induction of a magnetic field. The common perception was that some spin is responsible for the electrons magnetic field and for a moving electron it is enough to have a different velocity to generate a magnetic field for the observer.
But te thinking about

the movement of electrons around an atom ([which] would create a magnetic field)

and

electrons spinning around its axis ([which] generate a magnetic field)

could not be excepted as the reason for the magnetic field because simply nothing is rotating. Perhaps the magnetic dipole of electrons is a second intrinsic property (beside the electric charge)? And the spin is simply an expression for the behaviour of electrons in interaction with an external magnetic field?
Asking about the priority of electrons spin and of its magnetic dipole moment I’ve got an excellent answer that they are synonyms:


As regards the magnetic interactions of electrons, in particular, this means that nobody in the technical literature is going to say "intrinsic magnetic dipole moment" where saying "spin" will suffice. That's the case for electrons' intrinsic magnetic dipole moment, which is always proportional to their spin.
    This means that ... you need to get behind the usage of the term "spin" as synonymous with "intrinsic magnetic dipole moment", as in e.g. "spin-spin coupling" and "spin-spin interaction", ...


If your question concerns electrons as tiny magnets, the answer is yes, this should be a workable view.
