In my Computability and Complexity class, we are focusing on P, NP,
NP-complete, and NP-hard problems and the one thing that keeps coming up
is the SAT problem, in the context of reduction from one complexity to
another or to explain certain concepts.
Why is SAT so ubiquitous in theoretical computer science?

SAT was the first problem shown to be NP-complete, in Stephen Cook's seminal paper. Even nowadays, when introducing the theory of NP-completeness, the starting point is usually the NP-completeness of SAT.
SAT is also amenable to surprisingly successful heuristic algorithms, implemented by software known as SAT solvers. As a result, there is a lot of practical interest into formulating problems efficiently as instances of SAT.
SAT also shows up in fine-grained complexity, one of whose main assumptions is the strong exponential time hypothesis, which is a conjecture on the computational complexity of SAT.
