How would you define kinematic viscosity? What does it physically represent? Around the Internet I've found it defined as just a ratio, and that's it.
I saw in an answer that I can think of it as "diffusivity of momentum". What does this mean?
Answer
In mechanics, "kinematics" means describing the motion mathematically, so, for example, if the acceleration is known I can integrate to find the velocity and position. "Dynamics" means analyzing motion due to the influence of forces. The two are related through Newton's second law: F = ma (dynamics version) is the same as a = F/m (useful for kinematics).
In both cases, "coefficient of viscosity" refers to the effect on one part of the flow due to neighboring flow with a different speed, i.e. flow shear. That is, if you imagine yourself as one parcel of fluid and the next parcel over is moving in the same direction as you but faster, it will pull you along due to the effect of viscosity. "Dynamic viscosity" gives the force on your parcel (per unit volume), while "kinematic viscosity" gives the acceleration (force per unit mass).
(Diffusivity of momentum is also a valid way to think of it that is equivalent to this, but if you don't have any intuition about diffusivity to begin with that's not very helpful.)
ADDENDUM: Air and water make an interesting comparison: air has much lower dynamic viscosity than water (by a factor of 50), but due to its low density, it has much higher kinematic viscosity (by a factor of 17):
$\mu_\text{air} = 2\times 10^{-5}$ Pa s; $\;\mu_\text{water} = 1\times 10^{-3}$ Pa s
$\nu_\text{air} = 0.17 \text{cm}^2/\text{s}$; $\;\nu_\text{water} = 0.01 \text{cm}^2/\text{s}$
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