Dynamic viscosity (mu) is the real type of viscosity. It may also be called absolute viscosity as an interchangeable term. Dynamic viscosity is defined such that it is the proportionality constant when relating the shear stress the fluid exerts on the wall to the local gradient (derivative relative to a spatial coordinate) of the velocity profile (velocity as a function of space).
Dynamic viscosity has the units of Pascal-seconds, when used in the SI system.
Kinematic viscosity (nu) is a mathematical construct to make for less writing when solving numerous equations in fluid mechanics. It is by definition, dynamic viscosity divided by density (mu = nu/rho). You could also say "Greek m equals Greek n/Greek r". The reason it exists is that mu/rho appears frequently in the solving of the differential equations governing fluid flow.
Kinematic viscosity has the units of meters^2/second. This also matches the units of thermal diffusivity and mass diffusion coefficients. Often when studying generalized transport phenomena, it is useful to compare the three types of diffusivity (kinematic viscosity being one of them). In heat transfer, since convective heat transfer involves both study of flow and study of conduction, the Prandtl number is defined off of kinematic viscosity and thermal diffusivity.
Unfortunately, there exist names of the Poise and the Stoke respectively for both types of viscosity, but these were defined to be compliant with the obsolete centimeter-gram-second unit system. No one ever chose to make updated names for the two units. No one updated it, because most viscosities are so small that the cgs system is a more intuitive measure. Very seldom is viscosity ever an entire Pascal-second or an entire meter^2/second.