What is the moment of inertia really? Is moment of inertia or second moment of inertia, simply the resistance of a body to rotate it over an axis? What is radius of gyration? What if the axis is via the center of mass or somewhere different? can you give me please an overview of these issues with SIMPLE words, and without nonsense, like maths who nobody will ever remember. I need the SENSE how the brain comprehends these stuff in simple terms.
 A: 
simply the resistance of a body to rotate it over an axis?

Gosh, I dislike the word resistance in this context since resistance is, in general, dissipative and, in particular, resistance to rotation would imply that an isolated object that is rotating would eventually stop.
Think of moment of inertia (rotational inertia) about an axis as a measure of an object's opposition to change in rotation (about that axis) not as a resistance to rotation itself.
A: Basically, it is how hard it is to spin an object.  If you know what regular inertia is, moment of inertia is the rotation equivalent of it.
Regular inertia is how hard it is to push an object, and only depends on its mass.  For instance, picture an ice rink with a hockey puck.  If it is very light, then it is very easy to push it across.  However, for a very heavy puck, it will be hard to push it across.
Similarly, rotational inertia depends on its mass, but it also depends on its shape and where the it rotates.  Imagine spinning a top. It should spin in nice circles.  If you glue some stuff onto it unevenly, it will be harder to spin because you have changed the shape of it, increasing the moment of inertia.
Imagine a well shaped wheel.  It will rotate nicely.  If you put axle of the wheel somewhere other than the center, it will be harder to turn, because you increased its moment of inertia.  
A: The radius of gyration is roughly the proportion of the moment of inertia of a body to its mass.  A higher gyradius means that one body will have a larger moment of inertia than a body with a lower gyradius, if both bodies have equal mass.
A: The radius of gyration is the distance from the rotation axis you would have to squeeze the entire extended body onto and still maintain the same MMOI. 
So take any shape and change it into a thin cylinder around the axis of rotation. If the radius of the cylinder is the radius of gyration then both the shape and the cylinder have the same mass moment of inertia (provided they have the same mass).
