There are different definitions of acids and bases, each useful in their own contexts.
An early one is that of Arrhenius, who defined acids as molecules with a hydrogen "an H" and bases as molecules with a hydroxide "an OH". This definition works very well for elementary studies of strong acids and bases in water. It conforms to our expectations for acids like HCl, HNO3, HCN and others. It also rightly predicts basic behavior for NaOH, KOH. This definition was based on numerous observations which included how the chemicals behaved in water and how they reacted together. (One hates to imagine how many fingers he burned in the process.) Acids would make a glass of water sour, bases would make it bitter. Bases also tended to have a slippery feeling. When an acid and based were mixed, they would react to make water (H + OH) and a salt. (Salt in this case refers to an ionic compound, but not necessarily sodium chloride.)
We accept that Arrhenius' definition was correct, but it only accounts for certain chemicals. Over time it was found that there were other chemicals that had acid/base properties, but were not included in The Arrhenius definition. Two complementary theories were advanced.
Two chemists offered what we call the Bronstead-Lowry definition of acids and bases. This one defines an acid as a chemical species (may be a molecule or an ion) that can donate a hydrogen ion (also referred to as a proton). Bases are defined as chemical species that can accept a hydrogen ion. This model illustrates the reaction between an acid and a base as the passing of a hydrogen ion from an acid to a base. Under the Bronstead-Lowry definition, chemicals like ammonia (NH3) are considered bases, whereas they were not under the Arrhenius definition. This definition is a very good one for water-based chemistry.
Gilbert Lewis also provided a definition for acids and bases. His encompasses the previous two, as well as compounds and ions that the previous two do not. He defined an acid as a chemical species that can accept an electron pair and a base as a species that can donate an electron pair. By this definition, metals function as acids, the hydrogen ion itself is an acid, rather than being passed along by the acid, and electrons are the focus of chemistry, which is reflective of everything we know about chemical reactions. Antimony pentafluoride is an example of a molecule that is an acid only under Lewis' definition, yet it belongs to the inorganic acids that are much stronger than the aqueous acids dealt with in the other definitions. (SbCl5 easily eats through metal.)
It should be noted that under the last two definitions water is both an acid and a base. Such compounds are called "amphoteric". Another example of an amphoteric compound is an amino acid, which has a basic amine group on one end and a carboxyllic acid on the other.