I've studied a bit of propositional logic and first order logic, I know that propositional logic is sound and complete ($\Gamma \vdash \gamma$ if and only if $\Gamma \vDash \gamma$), I know what maximally consistent sets are, etc.
But I never understood how propositional logic could be used for anything, as in, what are some things we can represent with propositional formulas as to actually apply all the theoretical background.
For a well-known example of application (think at your cell-phone) see : Boolean algebra and Logic gates.
From the point of view of mathematical logic, propositional calculus is a "laboratory" where we can develop and learn the "machinery" - consistency, semantics, soundness, completeness, compactness - to be used with more complex languages and theories.