Look at the following theorem.
(Keisler–Shelah Theorem) Two $L$-structures $M$ and $N$ are elementarily equivalent if and only if there is an index set $I$ and an ultrafilter $D$ on $I$ such that $\big(\prod M\big)/D \cong \big(\prod N\big)/D$.
Keisler–Shelah Theorem is an interesting theorem for me but I have not seen any application of it.
Is it easy to use this theorem to show, for example, $(\mathbb{Q},<)$ and $(\mathbb{R},<)$ are elementarily equivalent?
As was pointed out in the comments, the direction "isomorphic ultrapowers -> elementarily equivalent" is really Łoś's theorem (and is much, much easier to prove than the converse).
So we should look for cases where the direction "elementarily equivalent -> isomorphic ultrapowers" can be used. To save some notation, let me write $M^{\mathcal{U}}$ for the ultrapower of a structure $M$ by the ultrafilter $\mathcal{U}$.
The applications I know of are typically of the following form: You're interested in some operation, let's call it $\sigma$, which turns structures of some kind into structures of another kind, and you want to show that if $M \equiv N$ then $\sigma(M) \equiv \sigma(N)$. You know that $\sigma$ commutes with ultrapowers, that is it has the property that $\sigma(M^{\mathcal{U}}) \cong (\sigma(M))^{\mathcal{U}}$. Then if $M \equiv N$ you have an ultrafilter $\mathcal{U}$ such that $M^{\mathcal{U}} \cong N^{\mathcal{U}}$, and therefore $(\sigma(M))^{\mathcal{U}} \cong (\sigma(N))^{\mathcal{U}}$, so $\sigma(M) \equiv \sigma(N)$.
A specific example of this type is when the original structures are fields, and you want to prove that if $F \equiv K$ then $GL_n(F) \equiv GL_n(K)$. Some details about this example, and links to more of this kind, are in the top answer to this MO post: https://mathoverflow.net/questions/158583/is-it-ever-a-good-idea-to-use-keisler-shelah-to-show-elementary-equivalence.