This project is a study of beginning first-order model theory, up to a point sufficient for an exploration of Vaught's theorem: there is no complete first-order theory with exactly two nonisomorphic countable models. The exploration of this theorem requires, in the first place, the knowledge of the notions of a first-order language, a complete theory, models and isomorphism of models; next, the understanding of theorems such as Completeness Theorem, Compactness Theorem, Lowenheim-Skolem-Tarski Theorem and Omitting Types Theorem. Lastly, atomic models and saturated models are also given a close look at after which these results would enable us to discuss the theorem by Vaught.