We have the following definitions and theorem related to NP-completeness.

Definition 1. Let $X_1$ and $X_2$ be two problems. $X_1$ polynomially reduces to $X_2$ (written as $X_1 \propto X_2$) if and only if $X_1$ can be solved in polynomial time, by using a polynomial time algorithm which solves $X_2$.

Definition 2. A problem is said to be a P (resp., NP) problem if it can be solved in polynomial time by a deterministic (resp., non-deterministic) algorithm.

Definition 3. NP (resp., P) is the set of all NP (resp., P) problems.

Definition 4. A problem X is an NP-complete (NPC) problem if $X \in NP$ and every NP problem reduces to X.

Cook's Theorem. $NP=P$ if and only if the satisfiability (SAT) problem is a P problem. (This implies that every NP problem polynomially reduces to SAT.)