Usually in logic correctness is a property of sequences/trees of formulas/sequents (depending on the system considered) that express they are valid proofs: that is that they are sequences/trees which are build applying the inference rules of the logic considered.
Soundness is a property of a logic, where by a logic we mean a proof system (i.e. a set of inference rules) and a semantics, basically a logic is sound if every time we have $T \vdash \varphi$ (that is $\varphi$ can be proved by the inference rules from the formulas in $T$) we also have that $T \models \varphi$ (that is $\varphi$ is true in every model of $T$).
So correctness is a concept that belongs to the syntactic setting while soundess links syntax with semantics.
Of course there are other possible use of the term correctness but these use belongs outside the realm of logic (or at least I cannot think of other uses of the word in logics).
Hope this helps.