What is Moment of Inertia and Centre of Gyration The center of mass is the point in a rigid body where all the mass can be concentrated and then the motion (translatory) of the body as a whole can be described from that point.
However can't say the same about the center of gyration as it changes with change in our choice of axis of rotation. So am I picturing it wrong? What is the right meaning of moment of inertia, and center of gyration? Visual representations are preferred please. Thanks. 
 A: Moment of inertia in simple sense means the resistance a body offers to any change that disturbs its state of rotation. It serves the same purpose as mass in non-rotational linear motion. More mass implies more inertia. Inertia just means resistance. 
For instance, when you calculate moment of inertia of a circular disc about its centre, you just add moments of inertia of all the infinitesimal masses that make up the disc. This gives you the total moment of inertia of the disc. 
$I=\frac{MR^2}{2}$ $\tag 1$
Now if you wish to replace this disc with a point mass, with the same mass as the disc, such that you get the same value of moment of inertia about the same axis, all you have to do is to put this point mass at such a distance from the axis that give you the same $MOI$.
Moment of inertia of a point mass is given by $mr^2$, where $r$ is the perpendicular distance of the point mass from the axis of rotation.
From the statements made earlier, 
$mr^2=(MOI)_{disc}$
$mr^2=\frac{MR^2}{2}$
$r^2=\frac{R^2}{2}$ $(\because m=M)$
$r=\frac{R}{\sqrt 2}$.
Here, $r$ is called the radius of gyration.  Don't confuse centre of gyration with the axis of rotation.
$MOI$ and radius of gyration change with axis of rotation, and they cannot be defined without a reference axis. $COM$ is simple a point mass with the mass of an object which gives you the same acceleration as the rigid object when acted upon by a force, irrespective of where the force is applied on the rigid body.
