Here is how I explained it to my mother, hopefully it will serve you :)
There are problems for which it is easy to find a solution (P, but less call them "easily solvable"), problems for which it is easy to check if a given solution is correct (NP, but let's call them "easily checkable"), and problems which are neither easily solvable nor easily checkable. For simplicity assume that "Easy" is formally defined, and that each problem has a unique solution.
Now, people have been able to prove (using mathematics) interesting relations between those two notions of "easily solvable" and "easily checkable", such that some problems are not easily solvable, and that some others are not easily checkable. A basic example of such result is that a problem which is easily solvable is also easily checkable: just find its solution and compare it to the solution given.
Tantalizingly enough, for a lot of practical problems (such as deciding if there is a possible assignment of students to professors and classrooms, when there is very little margin) it is not known if there is an "easy" way to solve it, but it is known how to check easily if a solution is correct or not. People tried a lot and failed, then tried to prove that it was not possible and failed as well: they just don't know. Some think that all problems which are easily checkable are easily solvable (we just should think more about it), some think the contrary, that we should not waste our time trying to find easy solutions to these problems.
What we found out is how to show links between problems (e.g. if you know how to go to school, you know how to go to the bakery which is just in front) and easily checkable problems which are linked to all other easily checkable problems (NP-complete, but let's call them "key problems") such that if someone, one day, shows that one of the key problems is easily solved, then all problems which are easily checkable are also easily solvable (i.e. P=NP). On the other hand, if someone show that one of the key problem cannot be easily solvable, then none of the others can be easily solvable either (i.e. P<>NP).
So the question is tantalizing, and relatively important in practice (although some argue that we should rather focus on alternate definitions of "easy"), and people are investing quite a lot of money and time in the debate.