In the 1870s, a young German mathematician named Georg Cantor began treating infinite collections as objects in their own right. Mathematicians had used the word "set" loosely for centuries, but Cantor proposed something audacious: the set of all natural numbers was itself a thing — a completed infinity — with definite properties that could be reasoned about and compared to other infinite sets. The result was set theory, a foundation so general that within fifty years almost every other branch of mathematics was being rewritten in its language. The price was a dose of paradox so sharp it nearly broke mathematics, and a kind of austerity that working mathematicians have alternately embraced and grumbled at ever since.
A set is a collection of distinct things; an element either belongs to a set or it does not, with no middle case. From that minimum, the rest is built: union, intersection, the empty set ∅, subset, the power set (the set of all subsets of a given set). Numbers themselves can be defined as sets — the empty set is 0; the set containing 0 is 1; the set containing 0 and 1 is 2; and so on. Functions are sets of ordered pairs. Algebraic structures — groups, rings, fields — are sets equipped with operations. Almost every modern mathematical object lives, formally, as a set with extra structure on it. The framework looked unassailable until 1901, when Bertrand Russell observed that the set of all sets that do not contain themselves either contains itself (in which case it shouldn't have) or doesn't (in which case it should have) — a paradox at the foundation. Mathematicians spent the next thirty years patching the system with axioms (Zermelo-Fraenkel, eventually with the Axiom of Choice — the system known as ZFC), restricting which collections counted as sets and which were merely "classes." The patched theory has held since. Cantor also discovered that not all infinities are the same size: the rationals are countable (can be matched one-to-one with the integers), the reals are not — the famous diagonal argument shows that no list of reals can include them all. Different infinities have different cardinalities, and the hierarchy goes up forever.
Set theory is the lingua franca underneath every undergraduate mathematics textbook, even when the textbook never names it. It also surfaces in databases (joins, projections, the relational model is essentially set algebra), in type systems in computer science, and in category theory's ongoing rivalry with set-theoretic foundations among working mathematicians. The deeper philosophical move — that mathematical objects can be defined by membership rather than construction — has migrated into philosophy of mind, ontology, and the formal semantics of natural language.