Reduction
Also known as: Many-one reduction, Polynomial-time reduction, Karp reduction
A reduction from problem $A$ to problem $B$ is a way to solve $A$ by translating instances into instances of $B$. If the translation is cheap and $B$ is easy, $A$ is easy too.
For NP theory the usual tool is a polynomial-time many-one (Karp) reduction: a poly-time function $f$ such that $x$ is a yes-instance of $A$ iff $f(x)$ is a yes-instance of $B$. Write $A \leq_p B$.
NP-hard means every problem in NP reduces to you. NP-complete means you are in NP and NP-hard. Reductions are how textbooks prove “this puzzle is as hard as SAT.”