There are some sorts of problems where it is relatively easy to check that a solution is correct, but hard to find that solution to begin with. For example, it is easy to check whether a large number is the product of two primes (429,496,729 = 19 X 22,605,091), but it is hard to find the factors of a large number. These problems are called ‘NP problems’ in mathematics, because they cannot be solved in polynomial time.
By a quirk of mathematics, if anyone ever comes up with an efficient way to solve one of these NP problems, the technique will be applicable to all such problems. That said, it may be the case that there is no efficient way to solve any of these problems. As such, whether all or none of them are efficiently soluble is an important question in mathematics.
The importance of the problem extends beyond the theoretical realm. For instance, the ‘traveling salesman’ problem is NP. There is a salesman who wants to visit X cities, with as little travel as possible. Finding the quickest route becomes dramatically more difficult as the number of cities increases. If someone could solve the P versus NP problem they could either help Fedex and UPS a lot (if an efficient solution to NP problems is found) or prove that their work will always be challenging (if it is proven that there are none).
This article provides more information on the P versus NP problem.