The Navier-Stokes equations describe one of the most ubiquitous features of our world: the flow of fluids like water. They are used to study ocean currents, the turbulence in the wave of an airplane or the flow of blood.
In addition, they are mathematically extremely interesting and there is a Millennium Prize Problem with a $1 million reward for the first person who can prove rigorously that solutions exist and answer the question if they are unique.
The Navier-Stokes equations are extremely important for modern transport. Not only designers of ships use them, but also aircraft and car engineers use it to make computer simulations to test the aerodynamics of objects.
In addition, the Navier-Stokes equation is used in medical research to calculate blood flow. This is essential, for example, during a coronary bypass operation, where the blood flow must be precisely controlled.
Another application is in climate models. These depend crucially on the flow of oceans and the atmosphere, which both can be modeled thanks to the Navier-Stokes equation.