Set
A set is a well-defined collection of distinct objects called elements, and is one of the most fundamental building blocks of mathematics.
Definition
A set is a collection of things grouped together; the things inside are called elements or members, and a set must be well-defined so that for any object, you can tell whether it belongs or not. Formally, a set $S$ is a collection of distinct, unordered objects, with membership denoted $x \in S$; two sets are equal if and only if they have exactly the same elements (the Axiom of Extensionality). In Zermelo-Fraenkel set theory (ZFC), sets are defined axiomatically, with key axioms including Extensionality, Pairing, Union, Power Set, and the Axiom of Choice; Russell's paradox (the set of all sets not containing themselves) showed naive set theory is inconsistent and motivated these axiomatic foundations.
Example
The set of vowels in English is $\{a, e, i, o, u\}$, using curly braces as the standard notation; is "b" in the set? No. Is "e"? Yes. Key number systems are built this way too: $\mathbb{N} = \{1, 2, 3, \ldots\}$, $\mathbb{Z} = \{\ldots, -1, 0, 1, \ldots\}$, $\mathbb{Q}$, and $\mathbb{R}$; order does not matter and duplicates collapse, so $\{1, 2, 3\} = \{3, 1, 2\}$ and $\{1, 1, 2\} = \{1, 2\}$. Russell's paradox asks: let $R = \{x : x \notin x\}$; is $R \in R$? Either answer leads to a contradiction, which is why ZFC restricts which collections can be sets.
Key Insight
Sets are the basic language of all of mathematics; almost every mathematical object, from numbers to functions, can be defined using sets. ZFC set theory is the standard foundation for modern mathematics, and independence results, like the Continuum Hypothesis being independent of ZFC, show there are mathematical questions ZFC cannot settle.