According to the work of Svante Arrhenius;

An acid is any substance which contributes hydrogen ions (H+) to the solution.

A base is any substance which contributes hydroxide ions (OH-) to the solution.

Svante August Arrhenius

(1859 – 1927)

This definition was the result of early research by Svante Arrhenius whose 1884 doctoral thesis concerned the dissociation of electrolytes. The research and analysis had ultimately earned him the Nobel Prize for chemistry in 1903.

Arrhenius' century-old definition is actually the definition one will get when one walks around asking random medical staff to define what an acid is.

The concept is reinforced by the education system which insists that

HA →  H+ + A-

This, according to at least one author, is a failure on the part of the education system. The definition is outdated and there is a good argument that it should no longer be used to poison the delicate minds of chemistry students. Though the concept was true for most of the acids and bases known at the time, even early in its history there was concern that as a definition it was unsatisfactory.

Problems with Arrhenius' definitions of acids and bases

Several issues with the definition must be acknowledged, so that it may meet with popular derision, and through ridicule become replaced by a more robust definition.

Hydrogen ions are largely imaginary.

Contrary to popular belief, hydrogen ions are not a real constituent of a solution. To be sure, an instantaneous snapshot of water may reveal some protons lurking between the molecules; however the proton affinity of H2O is massive - about 700 kJ/mol - which means the union of H+ and H2O (forming the "hydronium ion", H3O+) is massively favoured by the laws of thermodynamics. The proton is therefore a true rarity in an aqueous solution, as it is immediately absorbed. This was described by Lowry (of Brønsted-Lowry fame) as "an extreme reluctance of a hydrogen nucleus to lead an isolated existence". A team of chemistry teachers has posted a value estimating the true concentration of H+, which they report as being in the range of 10-130 moles/L.

The solvent must play a role.

Arrhenius' model does not take into account any solution other than the aqueous. The solvent is merely an unseen backdrop for the equations. However, obviously dissolving an acidic substance in a solvent with bizarre properties (eg. kerosene or liquid helium) will give rise to a solution with vasty different acidity when compared to water.

Salts dissociate into non-neutral solutions.

A salt dissolving in water should- according to Arrhenius - give rise to a perfectly neutral solution, as neither H+ nor OH- are being contributed. However, this is not the case; for instance salts that contain anions derived from weak acids form solutions that are basic.

References

The photographs of famous Nordic chemists are stolen directly from the Wikipedia articles concerning their great deeds, as these were labelled "for reuse". I presume this means the families of these chemists will not attempt to sue me for using their ancestor's likeness.

As for the history of acids and bases, apart from the lucid and comprehensive chapter from Kerry Brandis, one can turn to acid-base.com by Alan W. Grogono; the linked chapter draws heavily on the famous (in certain circles) book by Astrup and Severinghaus: The history of blood gases, acids, and bases ( Munksgaard, 1986). This opus is not available for free online, which is very sad (I'm looking at you, Google books).

Astrup and Severinghaus also wrote a series of review articles for the Journal of clinical monitoring, which are also locked up behind paywalls.

Severinghaus, John W., and Paul B. Astrup. "History of blood gas analysis. I. The development of electrochemistry." Journal of clinical monitoring 1.3 (1985): 180-192.

Severinghaus, John W., and Poul B. Astrup. "History of blood gas analysis. II. pH and acid-base balance measurements." Journal of clinical monitoring 1.4 (1985): 259-277.

Severinghaus, John W., and Poul B. Astrup. "History of blood gas analysis. III. Carbon dioxide tension." Journal of clinical monitoring 2.1 (1986): 60-73.

Severinghaus, John W., and Poul B. Astrup. "History of blood gas analysis. IV. Leland Clark's oxygen electrode." Journal of clinical monitoring 2.2 (1986): 125-139.

Severinghaus, John W., and Poul B. Astrup. "History of blood gas analysis. V. Oxygen measurement." Journal of clinical monitoring 2.3 (1986): 174-189.

Severinghaus, John W., and Poul B. Astrup. "History of blood gas analysis. VI. Oximetry." Journal of clinical monitoring 2.4 (1986): 270-288.

Severinghaus, John W., and Yoshiyuki Honda. "History of blood gas analysis. VII. Pulse oximetry." Journal of clinical monitoring 3.2 (1987): 135-138.

Buck, R. P., et al. "Measurement of pH. Definition, standards, and procedures (IUPAC Recommendations 2002). " Pure and applied chemistry 74.11 (2002): 2169-2200.

Arrhenius, Svante. Über die Dissociation der in Wasser gelösten Stoffe. Verlag von Wilhelm Engelmann, 1887. -The link points to a translation of the original paper.

Stephen J. Hawkes "Arrhenius confuses students." Journal of Chemical Education 69, no. 7 (1992): 542.

Peterson, Kirk A., et al. "Predicting the Proton Affinities of H2O and NH3." The Journal of Physical Chemistry A 102.14 (1998): 2449-2454.

Spektorowski, Alberto, and Elisabet Mizrachi. "Eugenics and the welfare state in Sweden: The politics of social margins and the idea of a productive society."Journal of Contemporary History 39.3 (2004): 333-352.