In the Brønsted–Lowry acid–base theory, an acid is defined as a proton donor and a base is defined as a proton acceptor. Here, proton means a hydrogen ion, H+ (a hydrogen atom without its electron is just a proton).
Some substances, such are water, are amphoteric (or amphiprotic). This means they are capable of acting as an acid or as a base in different chemical reactions. Acids can be classified as monoprotic (having one ionizable proton, like HCl) or polyprotic (having multiple ionizable protons, like the diprotic acid H2SO4).
In the following reaction, HCl(aq) is the acid and H2O(l) is the base because the former gives its proton to the latter:
HCl(aq) + H2O(l) → Cl−(aq) + H3O+(aq).
In the reverse reaction, Cl−(aq) would be the base, so it is called the conjugate base. Similarly, H3O+(aq) would be the acid in the reverse reaction, so it is the conjugate acid. A conjugate acid–base pair is a pair of substances whose formulas differ by a single proton. In the above reaction, the conjugate acid–base pairs are HCl/Cl− and H2O/H3O+.