indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
The Keisler-Shelah isomorphism theorem characterizes the partially syntactic concept of elementary equivalence in purely semantic form with the help of ultrapowers.
In practice, the theorem is usually stated as follows: let $A$ and $B$ be first-order $\mathcal{L}$-structures. Then $A$ and $B$ are elementarily equivalent, written $A \equiv B$, if and only if there is an ultrafilter $\mathcal{U}$ on some index set $I$ such that there is an isomorphism of ultrapowers $A^{\mathcal{U}} \simeq B^{\mathcal{U}}$.
Keisler proved, assuming GCH, that when $A \models T$ and $B \models T$ have cardinality $\leq 2^{|T|}$ then they have isomorphic $|T|$-indexed ultrapowers. Shelah removed the assumption of GCH at the cost of exhibiting the isomorphism for only $2^{|T|}$-indexed ultrapowers instead.
are elementarily equivalent. Assuming the continuum hypothesis (this is an example of where this technical distinction is vital), they are also isomorphic.
Los ultraproduct theorem?