Computability asks whether a problem can be solved. Complexity asks whether it can be solved efficiently — in time polynomial in the input size.
Definition · P and NP
P is the class of languages decidable in polynomial time on a deterministic TM. NPis the class whose “yes” answers can be verified in polynomial time given a certificate.
P ⊆ NP · (P = NP?) is openSudoku, the traveling salesman decision problem, and Boolean satisfiability all live in NP: a proposed solution is easy to check, even if finding one seems hard.