When an acid neutralizes a base (or vice versa) it forms a "salt", and water (typically we specify a water solvent when we speak of "acid-base chemistry").
Consider the simplest acid-base reaction:
NaOH(aq) + HCl(aq) rarr NaCl(aq) + H_2O(l)
Certainly, sodium chloride, unquestionably a salt (a binary compound of a metal and a non-metal), AND water have been formed as the result of neutralization of HO^- and H^+. I write (aq) beside the products and reactants, because in solution these species are ionized, e.g. HO^-, Cl^-, Na^+ etc.
The acidium species is often represented as H_3O^+; this again is a representation. As fas as anyone knows this is a cluster of water molecules, with an extra H^+, i.e. H_5O_2^+, H_7O_3^+. If we treat it as H^+, or H_3O^+, we are able to do calculations and stoichiometry.
All (aqueous) acid base chemistry can be represented by the word equation:
"Acid + base " rarr" salt + water"
