1. A decision problem, put simply, is a computational problem that can be posed as a yes/no question, based on a given set of input values. For example, it could be whether a given natural number is prime.
(ref)
2. A decision problem is decidable if there is an effective method, or a finite-time, deterministic procedure to solve it. This means there is a finite number of definitive instructions, always terminates after a finite number of steps, and always produces a correct answer.
3. P is the class of decision problems that can be solved in polynomial time. This means, the relation between the time (or the number of steps) it takes to solve a decision problem and the amount of input would be T=O(nk), where T is time, n is amount of input and k is a constant. Polynomial time is the most efficient complexity class and can be solved by deterministic machines, such as a computer. Meanwhile, NP class is the set of decision problems that are verifiable in polynomial time by a deterministic machine.
(ref1, ref2, and, ashamedly, ref3)
4. The P versus NP problem asks whether or not every problem that is verifiable in polynomial time is also solvable in polynomial time.
5. P versus NP is a Millennium Prize problem, so the Clay Mathematics Institute would reward 1 million US dollars (!!) to the first correct solver of the problem.