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The Compactness Theorem asserts that an infinite set Σ of first-order sentences has a model if and only if every finite subset has a model. This reduces satisfiability of infinite sentence sets to finite approximations, enabling construction of infinite models with prescribed properties through careful finite control.
From your study of first-order logic and the completeness theorem, you know that a set of sentences Σ is satisfiable if and only if it is consistent — has no finite proof of a contradiction. The Compactness Theorem is a direct corollary: a proof only invokes finitely many premises, so if every finite subset of Σ is consistent (satisfiable), then no finite proof from Σ can derive a contradiction, so Σ itself must be satisfiable. The heart of compactness is that first-order logic is blind to infinite sets of sentences — consistency is always witnessed by finite evidence.
The theorem's real power is in what it lets you *build*. The canonical application: suppose you want a model of the natural numbers that contains an element larger than every standard natural number. Take Σ to be the usual axioms of arithmetic, then add a new constant symbol c along with the infinite family of sentences {c > 0, c > 1, c > 2, c > 3, ...}. Any finite subset only requires c to exceed finitely many standard naturals, which is satisfiable (take c to be any sufficiently large standard number). Compactness then guarantees a model of the whole Σ — a non-standard model of arithmetic where c is infinitely large. No standard model satisfies this, yet the non-standard model exists and satisfies every first-order sentence true in ℕ.
This construction pattern appears repeatedly in model theory: to build a model with some "infinite" property, express it as an infinite set of first-order sentences, verify that every finite approximation is satisfiable, and invoke compactness to get the full model. The method works even when you cannot explicitly describe the model — compactness is an existence theorem, not a construction. Similarly, compactness shows that no first-order theory can *characterize* an infinite structure up to isomorphism: if a theory has any infinite model, it has models of all infinite cardinalities (by the Löwenheim-Skolem theorems, which themselves use compactness).
A complementary use of compactness is in proving non-expressibility results: if a property P cannot be approximated finitely (every finite approximation is satisfiable both by P-structures and non-P-structures), then no first-order sentence can express P. For example, "the domain is infinite" is not expressible by a single first-order sentence — you can express "there are at least n elements" for each finite n, but no finite sentence can force infinitely many. Compactness makes this precise: the union of finite-model sentences with "there are infinitely many elements" is satisfiable (every finite subset is), so the two properties cannot be separated by a first-order sentence. These non-expressibility results reveal the genuine limits of first-order logic compared to second-order logic or infinitary logic.