In very simple terms, combustion is the process of burning something.
More specifically, it is a chemical reaction between substances, usually including oxygen and usually accompanied by the generation of heat. In addition to heat, it's also common (although not necessary) for a combustion reaction to release light and produce a flame. The rate or speed at which the reactants combine is high, in part because of the nature of the chemical reaction itself and in part because more energy is generated than can escape into the surrounding medium, with the result that the temperature of the reactants is raised to accelerate the reaction even more. A familiar example is a lighted match. When a match is struck, friction heats the head to a temperature at which the chemicals react and generate more heat than can escape into the air, and they burn with a flame.