What is magnetic conductivity? I searched google for the meaning of magnetic conductivity but couldn't figure out what is it? 
electric conductivity is usually means that there is the electric field parallel to the interface is continuous between to interfaces, what about magnetic conductivity material ?
 A: This is what they told us about magnetic circuits and magnetic resistence at engineering school.
Magnetic reluctance (resistance) is similar to the concept of resistivity of simple resistors. Magnetic relucance is used when calculating a magnetic circuit e.g. transformer magnetic core, electromotor magnetic core, generator, etc.
Magnetic conductivity is the inverse of magnetic reluctance and resembles the electrical quantity conductance.
Here is a pretty picture of the analogy of electrical and magnetic circuits. By the way, the cite I took the picture off seems quite informative. 

Let us have an external magnetic field H of uniform intensity. That can be generated by a moving permanent magnet or by current through a loop.
The magnetic field density (magnetic induction) inside materials, introduced in the field is
B = μ0 * μR * H
Now let the magnetic flux be the magnetic flux density, multiplied by the cross-section of the magnetic circuit, introduced in the field. For simple math this cross-section has to be constant. So, divide the circuit in a sequence of equal-magnetic-permeability-equal-cross-section sections and model those as series resistors.
Φ = B * S
Now we can define magnetic resistance - it says "In a magnetic circuit of uniform cross-section, put in a uniform external magnetic field, there exists a scalar constant Rm that is the ratio between the applied external field and the resultant field strength within the material".
Rm = L / (μ * S) 
Example: calculate a simple transformer core
TODO
A: In systems where quantum interference effects play an important role due to a very large dephasing time, the conductivity $\sigma$ which you would get from classical physics is reduced due to constructive interference of electron loops in your sample. This is called weak localization. Applying an magnetic field B, which indeed breaks time reversal symmetry, changes the interference and can enhance the conductivity. $\sigma(B)-\sigma(B=0)$ is called magnetoconductivity.
A: There is no magnetic conductivity,since there is no real magnetic current. If there is real magnetic current, then magnetic conductivity is defined as the ratio of magnetic current density to magnetic field intensity - ie, magnetic conductivity = $M/H$
A: How about $\mu_r$? As in $\textbf{B} = \mu \textbf{H}$. In other words, for a given magnetic flux, $\textbf{B}$, the magnetic field strength, $\textbf{H}$, increases linearly with $\mu$.
