Here's how I would describe this to a non-techie. Since you have a tech background, I hope you can use this as a starting point for diving into this interesting, but very deep, question.
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Is P equal to NP?
We built computers to solve tedious problems that we lazy humans don't want to do by hand.
Some problems are pretty easy for computers to solve. For example, if you ask a computer to look page by page through a book (e.g., Atlas Shrugged) to search for a particular sentence, it can do it blazingly fast. Even if you double the length of the book, the computer will only have to perform twice as much work (still blazingly fast).
We'll call all of these "easy" problems P (think: "easy peasy").
Other problems are a bit harder for our current computers to solve. If we ask a computer to guess my password (20 characters long), it'll take a while before it arrives at the right answer. But if you saw me write down my 20-character password on a post-it, it would be very easy for you (and the computer) to check. All you'd need to do is try to log in! We'll call all of these problems, which are possibly hard to solve, but very easy to check, NP problems.
So the question we don't know the answer to:
Are all the problems in NP actually solvable in an "easy peasy" way? That is, are the set of all problems in NP just the same as the set of all problems in P?
We don't know! And a lot of smart mathematicians are working on the answer. If P=NP, that means "hard problems" are actually "easy problems." This might mean that my computer password would be more easily guessed... and a bunch of other ramifications.
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This reddit post was linked to earlier: http://www.reddit.com/r/explainlikeimfive/comments/mfswi/eli...
I thought it was a pretty good explanation (for me), so I hope you'll give it a chance.