Navier-Stokes is essentially F=ma in differential form for a fluid.
They are difficult to solve mainly because they are non-linear differential equations where the evolution of the solution at any one place&instant depends upon the solution at (just about) every other place&(past) instant.
To elaborate just a little, they are mass-momentum-and energy written in differential (or sometimes integral) form. They are all coupled meaning that you cannot under normal circumstances solve one, they must be solved simultaneously.
For different types of flows, they are dependent on "solutions" at certain points. For example, for supersonic flow, the equations become hyperbolic partial differential equations. This means that information only travels downstream. Luckily for CFD people, this means that the solution at one point only depends on a "cone of dependence", or points upstream.