The Arrhenius definition is the everyday one. It says an acid increases the concentration of hydrogen ions when it dissolves in water. The Bronsted-Lowry definition is broader.
It defines an acid as any substance that donates a proton to another substance, called a base. This donation can happen even outside of water. The Lewis definition is broader still.
It defines an acid as any species that accepts a pair of electrons. This covers reactions that do not involve protons at all. Acids share recognizable properties.
They taste sour, turn blue litmus paper red, and conduct electricity in solution. They also react with active metals to release hydrogen gas. Strong acids, like hydrochloric acid and sulfuric acid, ionize almost completely in water.
Nearly every molecule releases its proton. Weak acids, like acetic acid, ionize only partially, leaving most molecules intact in solution. This difference in ionization, not concentration, is what makes an acid strong or weak.
A common misconception is that a strong acid is simply a very concentrated one. Concentration describes how much acid is dissolved. Strength describes what fraction of it ionizes.
A dilute strong acid can still be more corrosive than a concentrated weak acid. When an acid reacts with a base, the two neutralize each other, forming a salt and water.