Intensional and extensional definition of a set

Two complementary ways of specifying a set (or a concept more generally). The distinction lies in whether you define something by its meaning/definition or by its contents/values.

Intensional (meaning/definition): defines a set or type by the rule, condition, or property that describes it. It cares how a set or concept is specified. $A = {x \in X ∣ P (x)}$
Extensional (contents/values): defines a set purely by listing its actual members. It cares only what is inside the set, regardless of how it was defined. $B = {a_{1}, a_{2}, \dots, a_{n}}$

Remarks

Two different intensional definitions may — and often do — produce the same extension. In classical set theory, a set is its extension: the axiom of extensionality asserts that sets with the same elements are equal. Thus intension determines extension, but not conversely: the extension does not recover the rule that produced it.
Intensional definitions are indispensable for infinite sets, which cannot be enumerated extensionally. Finite sets admit both.
The principle $P (x)$ is often called the defining property (in Spanish, definir por comprensión vs. definir por extensión).
The intensional/extensional dichotomy pervades mathematics and logic: type theory (intensional vs. extensional equality), functional extensionality, and the philosophy of language (Frege's Sinn vs. Bedeutung) all echo it. See Martin-Lof Type Theory for the type-theoretic version, where extensionality appears as a judgemental equality rule for functions.
The axiom of extensionality belongs to the axioms of Zermelo--Fraenkel set theory.