There are three different theories that have been put forth to define acids and bases - the Arrhenius theory, the Bronsted-Lowry theory, and the Lewis theory of acids and bases.

The Arrhenius theory

The Arrhenius theory of acids and bases states that an acid generates H+ ions when dissolved in water whereas a base produces an OH– ions when dissolved in water. 

TheBronsted-Lowry theory

The Bronsted-Lowry theory is based on a specific donor-acceptor model. This theory defines an acid as a proton (H+) donor and a base as a proton (H+) acceptor. This is a broader definition and one that is generally used in general chemistry classes. 

Lewis Theory

The Lewis theory is also based on the donor-acceptor model but it has a broader definition. This theory defines an acid as an electron pair acceptor or an electrophile, and a base is defined as an electron pair donor or a nucleophile.