First of all, viscosity is the resistance provided by one layer of a fluid to the other layer (layers in transverse direction to the velocity vector) when both are in motion relative to each other.
Now suppose two fluids with the same viscosity but different densities are allowed to fall freely under the action of gravity from the small diameter vertical tube. Certainly the fluid with higher density will fall more distance quickly, because of its higher weight (gravitational pull).
Again this time if we take those two fluids of different viscosities and allow them to fall through the tube, it becomes difficult for us to tell which fluid will fall more distance quickly.
This time both the viscosities and the density is different. So, to solve this problem we take the ratio of viscosity and density of a fluid and compare it. this ratio is known as kinematic viscosity.
The fluid with less kinematic viscosity will fall more distance quickly. Interestingly, the unit of kinematic viscosity is same as that of the acceleration. So the kinematic viscosity represents the negative acceleration offered to the flow.
Kinematic viscosity is also known as the momentum diffusivity. This is the ratio of the viscous force to the inertia force. Inertia means the state in which the body wants to be (or you can say inertia is the tendency of the body to resist the change of its state). In the above example the tendency of the fluid in the tube is to fall freely under the action of gravity. So its inertia will try to oppose any force which opposes the free fall (or the inertia will try that the fluid keeps on flowing).
But the viscosity will resist the flow. So its the relative strength of inertia and viscosity which will decide the net acceleration of fluid. This relative strength of viscosity and inertia is represented by kinematic viscosity.