We give an introduction to the theory of elementary submodels including the Tarski-Vaught criterion, Skolem hulls, and the downward Lowenheim-Skolem theorem. We will apply this theory to give a proof of the Delta System Lemma, a common lemma used in Set Theory. Time allowing, we will also use the theory of elementary submodels to give a proof of the fact that every first countable, countably compact, Hausdorff space X has size less than the continuum.