The extinguishing agents that can be found inside fire extinguishers can be divided into three categories, based on their extinguishing action: suffocation, heat reduction and chemical inhibition. Usually these actions can be combined.
An extinguishing agent suffocates when it prevents any contact between fuel and combustive agent. Inert gases, like nitrogen and carbon dioxide, and all dry chemicals and foams can be listed as extinguishing agents that suffocate fire.
Cooling the fuel is a different action: the extinguishing agent removes heat from the fuel so that its temperature goes below the ignition temperature. The main extinguishing agent that works this way is water, but also dry ice is worth mentioning, with its -78.5°C temperature.
If the extinguishing agent inhibits the chain reaction that leads to combustion by directly interacting with fuel, then we have a chemical reaction. In particular, we have a negative catalysis. Dry chemicals are an example of agents that extinguish fire through a chemical reaction. Until 1996 also halons (halogenated hydrocarbons) fell within this category. Since they were recognized as highly harmful to the ozone layer, they have been banned from the market.