>>12269445A basic computational problem can be thought of as the following, each of which is equivalent:
1) A decision problem to decide whether an input belongs to a set
2) Search for a specific subset given input
There are more types of problems, but these two are the basic ones. Look up what the properties of a Turing Machine are.
The important part in this discussion is decidability and determinism. A problem is decidable if and only if there exists a Turing Machine that successfully solves the problem and halts in finite time on every possible input. A Turing Machine is deterministic if every time it runs on the same input, it takes the exact same steps to solve it. You can think of determinism as the machine never having to make a choice to solve a problem.
P is the class of decision problems decidable by a deterministic Turing Machine in steps that are at most polynomial relative to the size of the input. NP is the class of problem for which given a problem L, and a proposed solution solution x, there exists a polynomially sized (in terms of x) certificate p such that there exists a machine V such that V takes L, x, and p as input, and can tell you if x is a solution in polynomial time. You can think of p as being a proof that convinces x that it solves L in a very short amount of time.
So informally speaking, P vs. NP asks if problems that are quick to solve are the same as the problems that are quick to verify solutions for.