In simple terms, a decision problem is a question that has a yes-no answer. When given an algorithm and a question paired with it the decision here tells if the solution is possible given a certain circumstance with possibly variables, note that the inputs or variables can be from an infinite set. These types of problems typically are seen in mathematical questions of decidability. Decidability determines if a decision problem is ‘decidable’ if there is an efficient and effective method to derive the correct answer.
Class P refers to an algorithm’s solving time complexity being of polynomial time. Whereas in contrast class NP refers to an algorithm’s verifiable time being of polynomial time. “NP” stands for ‘nondeterministic polynomial time,’ where nondeterministic means that whenever a random input is given into the algorithm, there can be a different behavior observed. Using this definition of P and NP, the idea of “P versus NP” states that if an algorithm is of class P, it is also of class NP.
This question is known to be one of the millennium prize problems which is that if you are the first to solve it then you are awarded one million dollars as a prize. However there is a theory/rumor that says if it is proven that P=NP then you should theoretically be able to hack into the world’s secure banks. (Though it being possible and achievable are two different things).