P vs NP is a million dollar millenium problem.
Essentially it boils down to, If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.
Per a comment, it became clear that I should add some extra clarity! :)
In what ways can we intuitively teach HS-college students how to understand and attack a problem like this. Not P vs NP in particular but rather, a truly difficult, if not unsolvable problem. I think finding ways to to reconcile these types of problems, is important to developing really fine tuned critical thinking and analysis skills. With that, I do want them to have an intuition around P vs NP in this particular case, to set the stage for a very difficult problem. And I do want them to try out ways to approach difficult problems