Q:

Can you explain Navier-Stokes equations to a layman? 

Can you explain Navier-Stokes equations to a layman?  Could someone explain this famous and important equation with "plain words"?  If my question is too broad for an answer, I will also be very thankful for some introductory words.  

A:

This is my view of some physical phenomena that The Navier Stokes equations can be applied to, and why they are so complicated.  
It centers around oddness, counter intuitive behaviour of physical systems that we are trying to model using mathematics.  
Odd Behaviour #1 
If you measure the friction introduced  into a fluid as you push it through  a tube, you will find that, counter-intuitively, as the speed of the fluid increases, the friction reduces in magnitude.  Then the friction increases again as you speed up the water and finally it settles at a constant value. 
Odd Behaviour #2
Look over a bridge and watch the turbulence of the water around obstacles such as rocks or bridge supports.  This flow looks, but isn't totally random, it is actually less random than the surface of a "calm" lake.  The molecules of the lake water surface are not connected / correlated with each other, they move in random direction.  But if you look long enough at turbulent water, patterns appear, last for a while, then other flow patterns replace them. 
Why is there turbulence and eddies and why do they act this way when they flow past a rock, even a streamlined one that has been worn down to a smooth shape? 
One idea is that we can attempt a math solution by assuming that, as in the friction example above, some physical quantity gets maximised, and that although it looks a mess of different flows, it is actually the most efficient method of travel for the water molecules. 
Odd Behaviour #3
When you create a magnet, say by passing a permanent feromagnetic over a piece of steel, the steel acquires it's magnetic properties as the discrete magnetic domains start to line up, the usually illustration is of a large arrow created by many small arrows all "following the herd".  
This magnetic effect will fade over time, but you can also get rid of it by heat the steel bar and imparting random motions to the domains, so their magnetic fields cancel each other out. 
But, as you raise the temperature, there is a certain temperature,  "the Curie temperature", where aligned clusters of domains can form, because each domain can still feel the magnetism of the atoms around it.  Each cluster points in a random direction. 
This is something like how eddies might form in turbulent water, if you compare the rise in temperature to the increase in speed of the fluid that flows through the tube. 
My point that all this physical behaviour is very confusing and elaborate, compared to  "ordinary" physical  behaviour, such as a mass simply  bouncing up and down  on a spring in a regular, predictable (boring) way. 
The equations describing odd behaviour   must be complicated,  precisely because it is non-intuitive.   Therefore, they must be difficult, sometimes very  difficult,  to solve.  

A:

Perhaps then this kind of answer is what you are looking for: The Navier-Stokes equations are simply an expression of Newton's Second Law for fluids, stating that mass times the acceleration of fluid particles is proportional to the forces acting on them.  If we take the Navier-Stokes equations for incompressible flow as an example, which we can write in the form
$$\rho\left(\frac{\partial\mathbf u}{\partial t}
+{\mathbf u}\cdot{\mathbf\nabla}{\mathbf u}\right)=-{\mathbf\nabla}p+\nu\Delta{\mathbf u}+{\mathbf f},$$
we can see that the left-hand side is the product of fluid density times the acceleration that particles in the flow are experiencing.  This term is analogous to the term $m\,a$, mass times acceleration, in simple statements of Newton's law.  The only minor difference is that here we have divided by the fluid volume to obtain a volume-specific mass.  On the right-hand side, then, we have the volume-specific forces that are responsible for particle acceleration: the pressure gradient, the effect of viscous shear stresses, and volume forces.  If the volume forces are due to gravity alone, then we have ${\mathbf f}=\rho\,{\mathbf g}$, where $\mathbf g$ stands for the gravitational acceleration vector. 
Hope this helps. 

A:

For the complete layman explanation, it's really very simple:

Mass cannot be created nor destroyed
$F = ma$, otherwise known as Newton's second law
Energy cannot be created nor destroyed.  

There you have it.  That's continuity, momentum, and energy equations.  All of the rest that is "complicated" comes down to defining the types of forces (pressure, viscous), defining the types of energy (internal/thermal, potential, transfer between them) and then defining your coordinate system and reference frame (Lagrangian or Eulerian). 
In other words, the layman is just like pretty much every other classical physical system. 