In 1923, Johannes Brønsted and Thomas Lowry, working independently, defined an acid as a particle that gives a hydrogen ion and a base as a particle that takes one.
When hydrogen chloride dissolves in water, each molecule hands its hydrogen ion to a water molecule: \mathrm{HCl + H_2O \longrightarrow H_3O^+ + Cl^-}. Hydrogen chloride is the acid, and water, which takes the hydrogen ion, is the base.
A hydrogen ion is a single proton, so an acid is a proton donor and a base is a proton acceptor. The reaction is the same one that Arrhenius described, but now both particles have a role.
When ammonia dissolves, a water molecule gives a hydrogen ion to an ammonia molecule. Water is the acid this time, and ammonia is the base.
The products are an ammonium ion, \mathrm{NH_4^+}, and a hydroxide ion, \mathrm{OH^-}, so the solution is basic. This explains why ammonia counts as a base even though it has no hydroxide ions of its own.
An ammonium ion can also give its hydrogen ion back to a hydroxide ion, so the reaction goes both ways. The double arrow in the figure shows this, and only a small part of the ammonia is turned into ions at any moment.
With hydrogen chloride, water takes a hydrogen ion and becomes an oxonium ion, \mathrm{H_3O^+}, so it acts as a base. With ammonia, water gives a hydrogen ion and becomes a hydroxide ion, so it acts as an acid.
Whether a particle is an acid or a base therefore depends on what it reacts with. A substance that can act either way is described as amphoteric, or more precisely amphiprotic, and water is the most important example.
Under the Arrhenius definition water is neither an acid nor a base, so this is one place where the newer definition says more.
Hydrogen chloride and ammonia also react as gases. Where the two gases meet, white smoke forms: \mathrm{NH_3 + HCl \longrightarrow NH_4Cl}.
Each hydrogen chloride molecule passes its hydrogen ion straight to an ammonia molecule. The ammonium ions and chloride ions attract each other and pack into tiny crystals of solid ammonium chloride, which form the smoke.
No water is present, so no hydroxide ions or oxonium ions form and the Arrhenius definition cannot describe this reaction. In the Brønsted-Lowry definition it is simply another transfer of a hydrogen ion from an acid to a base.
When an acid gives away a hydrogen ion, what remains can take a hydrogen ion back, so it is a base. This base is called the conjugate base of the acid, and the acid is the conjugate acid of the base.
In the reaction of hydrogen chloride with water, \mathrm{Cl^-} is the conjugate base of \mathrm{HCl}, and \mathrm{H_3O^+} is the conjugate acid of \mathrm{H_2O}. Each acid-base reaction contains two such pairs.
In the reaction of ammonia with water, the pairs are \mathrm{NH_4^+} and \mathrm{NH_3}, and \mathrm{H_2O} and \mathrm{OH^-}. The two members of a pair always differ by exactly one hydrogen ion.