What languages can be reduced to a NP-complete problem in polynomial time
Under Karp reductions, the answer is exactly $\mathbf{NP}$: it is not hard to see that if a language is Karp-reducible to any $\mathbf{NP}$-language, then it is in $\mathbf{NP}$ too. On the other hand, all of $\mathbf{NP}$ reduces to $\mathbf{NP}$-complete languages by definition.

Under Turing reductions, the answer is the class $\mathbf{P}^\mathbf{NP}$: languages that are decidable by a polynomial-time algorithm that has an access to an $\mathbf{NP}$ oracle. Indeed, a Turing reduction of a language $L$ to an $\mathbf{NP}$-complete language is by definition a polynomial-time algorithm deciding $L$ with oracle access to $\mathbf{NP}$. On the other hand, any such algorithm can be simulated with oracle access to an $\mathbf{NP}$-complete language, thus obtaining a Turing reduction.