There are actual several definitions of acids and bases. The simplest of these is the Arrhenius definion. An acid is a substance, such as HCl, that generates H+ ions in water. A base (also known as an alkali), such as NaOH, is a substance that generates OH- ions in water. Another common acid-base definition is Bronsted-Lowry which states that an acid is a proton donor and a base is a proton acceptor.  The Bronsted-Lowry definition is broader and more accurately represents acids and bases.  It also accounts for the fact that some substances, such as water can act as an acid or a base depending on what the other reactant is.  H2O can donate a proton when paired with a base such as NH3 and accept a proton when paired with an acid such as HCl.