One of my favourite applications is proving that $\sqrt2$ is irrational using ultraproducts.

This requires knowing a nontrivial fact, that there are infinitely many prime numbers $p$ such that $x^2\not\equiv 2\pmod p$ for all $x$. In other words, $\Bbb F_p$ does not have a solution for $x^2-2=0$. Once we know there are infinitely many such prime numbers, we can take the ultraproduct of these $\Bbb F_p$ to obtain a field of characteristics $0$ where $x^2-2=0$ does not have a solution, since $\Bbb Q$ is a subfield of that field, it is impossible that $x^2-2$ can be solved there, and therefore $\sqrt2$ is irrational.