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Morley's Categoricity Theorem states: if a countable theory is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. This major breakthrough suggests categoricity in high cardinalities forces highly constrained structure. The theorem motivates stability theory: countable theories categorical in some uncountable cardinality are provably ω-stable.
From your study of categorical theories, you know that a theory T is κ-categorical if it has exactly one model of cardinality κ up to isomorphism. The two cleanest examples at countable cardinalities are the theory of dense linear orders without endpoints (DLO), which is ℵ₀-categorical (by a back-and-forth argument), and the theory of algebraically closed fields of characteristic p (ACF_p), which is κ-categorical for every *uncountable* κ but not for ℵ₀. Morley's theorem says these two regimes are not independent: uncountable categoricity is an all-or-nothing affair for countable theories.
The theorem's content is that uncountable cardinalities are not independent witnesses to structure. If a countable first-order theory T is categorical in some uncountable cardinality κ, then T is categorical in *every* uncountable cardinality. The proof, which Morley gave in 1965 and which inaugurated modern model theory, works by showing that such theories have a very constrained type space: they are ω-stable, meaning that for every countable set of parameters A, the space of complete types over A is itself countable. ω-stability prevents the theory from having "too many" types over countable sets, and this rigidity propagates to force unique models at all uncountable cardinalities.
The key ingredient in the proof is Morley rank, a dimension-like ordinal assigned to definable sets. In an ω-stable theory, every nonempty definable set has a well-defined Morley rank — an ordinal that measures how "large" or "ramified" the set is. Two models of the same uncountable cardinality in a Morley-categorical theory turn out to have the same Morley rank everywhere, and a back-and-forth construction using this rank builds an isomorphism between them. The rank provides the structural rigidity that forces categoricity. In ACF_p, Morley rank coincides with Krull dimension of algebraic varieties — a satisfying connection between the abstract model-theoretic invariant and a classical geometric one.
Morley's theorem is a founding result of stability theory because it shows that categoricity forces ω-stability, and ω-stability is a well-defined algebraic property that can be studied on its own terms. Shelah's subsequent work generalized this: instead of asking "when is a theory categorical?" he asked "how many non-isomorphic models can a theory have in cardinality κ?" The answer turned out to depend on whether the theory is stable, superstable, ω-stable, or none of these — a hierarchy that Morley's theorem first suggested was the right one to study.