Well for mathematicians, we tend to think of ODEs/PDEs in terms of:

Given a system of ODEs/PDEs, and a set of Initial/Boundary conditions, does a solution exist? If it exists, is this solution unique? Is it possible to characterise this solution in some way, e.g. is it differentiable, or is it a smooth (infinitely differentiable) solution?

Example: You may remember in engineering maths classes (assuming you did it) that some boundary value problems for some simple ODEs did not have a unique solution.

Back to the Navier-Stokes equation: The challenge (in the Clay Math prize) is, given the Navier-Stokes PDEs, and a set of quite general initial conditions, show that a smooth solution exists globally (not locally), or show otherwise, that no such solution exists.

Interestingly enough, Perelman's proof of the Poincare Conjecture also uses a PDE, the Ricci flow, in which he showed that the singularities of the Ricci flow were 'well behaved' and disappeared in a finite time, giving the 'answer' to Thurston's geometrisation conjecture, which then implies Poincare's conjecture. [I'm only describing approximately what he did here, this is as far as my understanding goes]