Curry-Howard is the idea that static types are propositional formulas and expressions that have such types are their proofs. Under the Curry-Howard correspondence, a static typechecker analyzing your program is equivalent to a proof checker ensuring that your proof is valid.
People tend to invoke Curry-Howard to praise ML- and Haskell-style algebraic types (according to Curry-Howard, sum types are disjunction, product types are conjunction, and function/exponential types are implication).