Think of a decision problem as a yes/no question posed in terms of inputs. You give it some data (an “input”), and the question asks: does this input satisfy some property? If yes, answer “Yes”; otherwise “No”. For example: “Is this number prime?”, “Does this graph have a path visiting every node exactly once?”
A decision problem is decidable if there is some algorithm that always finishes (on every possible input) and correctly answers Yes or No. If no algorithm can do that (meaning some inputs either can't be handled or the algorithm never finishes in some cases), the problem is undecidable.
P is the set of decision problems for which there is an algorithm that solves them efficiently, meaning the time it takes is bounded by a polynomial in the size of the input. Roughly: if the input size is n, time might be something like n2, n3, etc. Those are tractable problems.
NP stands for “nondeterministic polynomial time.” A decision problem is in NP if whenever the answer is “Yes”, there's a short “certificate” or “proof” (something you can check) that you can verify efficiently (in polynomial time). In other words, once someone gives you a candidate solution, you can check it quickly. But finding it in the first place might be hard.
This is the big question: If you can check a solution quickly, does that mean you can also find a solution quickly?
If P = NP, then every problem whose “yes”-answers are easy to check would also be easy to solve from scratch. If P ≠ NP, then there are problems where you can verify a given solution fast, but no fast way is known (or believed possible) to find the solution in the first place.
If you prove it one way or the other, you get a $1,000,000 prize from the Clay Mathematics Institute. Also, your name will go into textbooks, you’ll get huge recognition in math / CS communities. It would be a major breakthrough with major implications (cryptography, optimization, algorithms, etc.).