There are 3 main types of air purifiers: Electronic air cleaners, dehumidifiers, and ultra violet germicidal lights.
Electronic Air Cleaners - Most use electricity to make the particles in the air be negatively or positively charged. Than another part of it is the magnet which will grab the particle that was just charged. These usually do a great job of cleaning the air but they have to be maintained very well to keep up the effectness.
Dehumidifiers - This controls the humidity in your house. If you have high humidity in your home, it causes things to grow on the particles that you breathe. If you run an air conditioner and a dehumidifier at the same time, the dehumidifers will keep the inside of the air conditioner dry which prevents things from growing inside.
Ultra Violet Germicidal Lights - These are supposed to use a light that kills stuff in the air but there are mixed reviews on these. But the air is moving too fast for the light to do anything.