The Arrhenius Theory of acids and bases
The theory
Acids are substances which produce hydrogen ions in solution.
Bases are substances which produce hydroxide ions in solution.
Neutralisation happens because hydrogen ions and hydroxide ions react to produce water.
Limitations of the theory
Hydrochloric acid is neutralised by both sodium hydroxide solution and ammonia solution. In both cases, you get a colourless solution which you can crystallise to get a white salt - either sodium chloride or ammonium chloride.
These are clearly very similar reactions. The full equations are:
In the sodium hydroxide case, hydrogen ions from the acid are reacting with hydroxide ions from the sodium hydroxide - in line with the Arrhenius theory.
However, in the ammonia case, there don't appear to be any hydroxide ions!
You can get around this by saying that the ammonia reacts with the water it is dissolved in to produce ammonium ions and hydroxide ions:
This is a reversible reaction, and in a typical dilute ammonia solution, about 99% of the ammonia remains as ammonia molecules. Nevertheless, there are hydroxide ions there, and we can squeeze this into the Arrhenius theory.
However, this same reaction also happens between ammonia gas and hydrogen chloride gas.
In this case, there aren't any hydrogen ions or hydroxide ions in solution - because there isn't any solution. The Arrhenius theory wouldn't count this as an acid-base reaction, despite the fact that it is producing the same product as when the two substances were in solution. That's silly!
The Bronsted-Lowry Theory of acids and bases
The theory
An acid is a proton (hydrogen ion) donor.
A base is a proton (hydrogen ion) acceptor.
The relationship between the Bronsted-Lowry theory and the Arrhenius theory
The Bronsted-Lowry theory doesn't go against the Arrhenius theory in any way - it just adds to it.
Hydroxide ions are still bases because they accept hydrogen ions from acids and form water.
An acid produces hydrogen ions in solution because it reacts with the water molecules by giving a proton to them.
When hydrogen chloride gas dissolves in water to produce hydrochloric acid, the hydrogen chloride molecule gives a proton (a hydrogen ion) to a water molecule. A co-ordinate (dative covalent) bond is formed between one of the lone pairs on the oxygen and the hydrogen from the HCl. Hydroxonium ions, H3O+, are produced.