A fire or flame typically occurs when combustible airborne chemicals (fuels) oxidize (or burn) in such a way that heat is released (exothermic) and the electron activity or the temperature (incandescence) of the reaction emits visible light. The fire is the volume in the air where the chemicals are reacting in this way and emitting light. So, I guess the best answer I can give is that fire is made of vigorous chemical reactions. Most fires also "feed" themselves because the heat produced by the flame drives more fuel particles from a solid or liquid source into the air. Note that it is possible to have an invisible "fire" if the fuel burns in a way that does not produce visible light. Some alcohols burn with invisible flame - but that is not what people typically imagine when they think of fire. This kind of fire is also the region/volume of vigorous exothermic chemical reaction. It is also possible to have fire without oxygen by combining other reactive chemicals, but in our atmosphere burning by oxidation is by far the most common phenomenon. It is also possible to have fire outside our atmosphere (no air), but the principles are essentially the same: mix stuff, chemical reaction makes heat and typically light.

It's made of burning gases.

In most cases heat Fuel and oxygen. If you remove any one you eliminate the fire. The exception to this rule are metals that burn without oxygen they must be deep-sixed.