Whenever acids & bases are mentioned, they're usually the Bronsted-Lowry type (H+
donors & acceptors). You might also hear about Lewis acids & bases, which involve donors/acceptors of electrons instead of H+
, but these will be refered to as "Lewis acids" or "Lewis bases" (or often in organic chemistry, "electrophiles" & "nucleophiles") rather than simply "acids" or "bases" (unless your chemistry teacher or textbook happens to be fiendishly ambiguous
Anions of strong acids--like HSO4-
--won't accept H+
, at least not to any significant degree. So, not all negative ions will act as a (Bronsted-Lowry) base.