There are three basic types of fire extinguishers. Each type of extinguisher may be rated for one or more classes of fire. In some cases, particular extinguishers are not only considered ineffective against certain classes of fire, they can be dangerous if used in those circumstances.
Firstly, you have the water-based fire extinguisher. These are filled with pressurised water, and are made to put out type A fires from combustible materials, such as rubbish, clothing, wood and papers.
Secondly, you have dry chemical extinguishers. Dry chemical fire extinguishers are by far the most common fire extinguishers in the home. They can handle up to three types of fires. Type A: combustible solids like wood or paper, Type B: combustible liquids like gasoline or grease, and Type E: electrical fires.
Thirdly, you have carbon dioxide fire extinguishers. A carbon-dioxide fire extinguisher works by eliminating oxygen and replacing it with carbon dioxide. This is most commonly used to extinguish type E fires, electrical fires.
In addition to these three, you also have foam extinguishers and wet chemical extinguishers.
These different types of extinguishers are not interchangeable. In fact using the wrong fire extinguisher could make matters a lot worse. For instance, using a water-based extinguisher on a type B fire will only make it worse. A dry chemical extinguisher will put out a type E fire, but it will leave a sticky or corrosive residue that can damage the surface on which it has been sprayed. Moreover, for Type F fires, cooking oil and fat fires, a fire blanket must be used and not a fire extinguisher. The pressurised force from a fire extinguisher onto burning oil and fat could blow and spread it and make it worse.
For most homes, it will be more than sufficient with a combination of dry chemical extinguishers and fire blankets in addition to smoke alarms.