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Does P Equal NP?
The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified (is in NP) can also be solved quickly (is in P).
The question is, "Are all NP problems really just P problems?"
The answer is, "We don't know, but probably not".
Why Do We Care?
All problems in NP (you know, hard ones like the traveling salesman problem) have been proven to also be solvable in polynomial time if we can find a solution to just one NP-Complete problem.
If a single NP-complete problem can be solved quickly (in polynomial time) that means that all problems in NP can be solved in polynomial time. That would be a huge deal, particularly because it would break digital security systems that rely on the difficulty of certain NP problems.