The ancient Greeks and alchemists thought that fire was an element. They also considered earth, air, and water to be elements. However, the modern definition of an element defines it by the number of protons a pure substance possesses. Fire is made up of many different substances, so it is not an element.
For the most part, fire is a mixture of hot gases. Flames are the result of a chemical reaction, primarily between oxygen in air and a fuel, such as wood or propane. In addition to other products, the reaction produces carbon dioxide, steam, light, and heat. If the flame is hot enough, the gases are ionized and become yet another state of matter: plasma.