Svante Arrhenius' theory of electrolytic dissociation was ill-received when he first published it---in fact, his 1884 dissertation on the topic was ranked only "fourth class," something akin to a D in the modern American grading system, which was certainly sorely disappointing. The importance of this contribution to chemistry slowly gained recognition, and in 1903, he was awarded the Nobel Prize in Chemistry expressly for the electrolytic dissociation theory.
Svante's theory was the first really modern theory of acids and bases; previous definitions tended to describe the compounds by content, and the formation of free, charged ions was not considered. He thus gave a wholly new method of understanding the properties and reactions of these compounds, and while several definitions of acids and bases have been advanced since, they have largely been refinements or additions to the Arrhenius definition. Thus the Arrhenius concept of electrolytic dissociation remains the foundation of modern acid-base theories.
It is this model that made the pH scale useful (it measures the concentration of H+ ions in a solution), and similarly the pOH scale (measuring the concentration of OH- ions). The Arrhenius definition greatly simplified the understanding of acidic and basic compounds, providing the first systematic explanation for their properties in solution, most notably the fact that both types conduct electricity when in solution. It also explained why the two types of compounds neutralize one another (the H+ and OH- ions combine to form neutral water molecules, and the other ions combine to form neutral salts).
The Arrhenius definition of acids and bases is used to describe the activity of certain compounds in solution. Most simply it accounts for the fact that, if an acid and a base are both dissolved into a solvent (namely water), their properties are neutralized, as a result of forming a salt and more water. As many salts remain ionized and in solution, and water already is the solvent, this is often a less than readily observable reaction. It is, nonetheless, an extremely useful depiction of the reactions, providing an explanation for their reactivity and many common properties.
While the Arrhenius definition of acids and bases revolutionized the understanding of these compounds' behaviors in solution and their reactions with one another, it was by no means a complete representation. At the time, one of the most notable problems was the inexplicability of ammonia (NH3)'s basic properties. Another problem with the definition is its lack of attention paid to the solvent that the compounds are dissolved in---ions that readily form in water do not break so easily, if at all, from their compound in other types of solvents. A third point would be that it is known today that free protons (the H+ that Arrhenius was so fond of) do not really occur in solution, tending instead to form hydronium ions (H3O+) when in water solution. H+ is still widely used as a shorthand when describing these reactions, however.