There are a few different definitions for an acid. The Arrhenius definition of an acid is any chemical that increases the proton concentration in solution. Protons are hydrogen cations (H+) but in aqueous solution they coordinate with water to form the hydronium ion (H3O+). The Bronsted-Lowry definition is that an acid is anything that is a proton donor. This is probably the most common definition of an acid. A proton donor can include everything from strong acids like HCl to weaker acids like acetic acid and even the protons from alcohols and amines which can be proton donors when treated with a base. The other definition of an acid is the Lewis definition which states that an acid is an electron pair acceptor. This includes all of the previous chemicals covered plus things like electron deficient metals like boron and aluminum that contain no protons yet can act as acids in chemical reactions.
*operational definition for an acid when added to water is pH < 7.