Arrhenius Definition of Acids and Bases
The pH scale is based on the assumption in order to measure acidity we simply need to measure the concentration of H+ ions in an aqueous solution. Conversely the pOH scale is based on the assumption that the strength of a base can be determined by measuring the concentration of OH- ions in an aqueous solution.
This model is known as the Arrhenius definition of acid-base reactions named for Svante Arrhenius who first proposed the model in 1884. Put succinctly.
“Arrhenius acids form hydrogen ions in aqueous solution with Arrhenius bases forming hydroxide ions”
Brønsted-Lowry Definition of Acids and Bases
When HCl gas reacts with NH3 gas a white solid known as Ammonium Chloride (NH4Cl) is formed.
Note that there are no free OH- or H+ ions involved in this reaction and that by the Arrhenius definition this could not be considered to be an acid / base reaction.
However, if the same reaction were to be carried out in an aqueus environment, it would be considered to be an acid base reaction. Clearly the Arrhenius definition needs expansion in order to take into account a wider range of reactions and reaction environments. One such expanded theory is referred to as the Brønsted-Lowry definition, after Johannes Nicolaus Brønsted and Martin Lowry who each independently proposed the model in 1923.
In the Brønsted-Lowry definition, acids, referred to as "Brønsted-Lowry acids" donate H+ to a reaction and "Brønsted-Lowry bases" accept H+ ions from a reaction. Since a positive hydrogen ion is normally a single proton nucleus with no accompanying electrons or neutrons. The shorthand way to refer the these types of acids and bases has become: "proton donors" for "Brønsted-Lowry acids"; and "proton acceptors" for "Brønsted-Lowry bases".
This new acid / base definition has the advantage of completely encompassing the older Arrhenius definition, while expanding the definition to include not only nonaqueous reaction environments, but whole new classes of reactions which involve the exchange of H+ ions (protons) between molecules.
In summary, if you are chemistry beginner, acids and bases can be safely thought of as a measure of the H+ and OH- ion concentrations in aqueous solutions. However, as you become an intermediate student of chemistry, you will require a more sophisticated understanding of reaction mechanisms and the Brønsted-Lowry definition of acids and bases will become a critical part of that understanding.
Lewis Definition of Acids and Bases
An even more general definition of acids and bases was published by Gilbert N. Lewis, also in 1923. Rather than characterise the reaction as the exchange of a proton or H+ ion, Lewis defined the acid base reaction as the donation or aceptance of an electron pair.
Like the Brønsted-Lowry model, the Lewis model describes reactions in nonaqueous environments however it also expands the definition of acids and bases to situations were H+ ions are not exchanged.
In the Lewis definition, an acid reacts with a base by forming a new covalent bond utilizing an empty orbital of the acid to share the extra electron pair of the base.
Understanding of the Lewis model will be required by those of you who go on to study advanced chemistry, whereas the Brønsted-Lowry definition will be sufficient to understand the most commonly observed acid / base reactions.
Wikipedia: Acid-base reaction theories
IB Chemistry Web: 18.1 Brønsted-Lowry Theory
IB Chemistry Web: 18.2 Lewis Theory