Bacteria: Airborne bacteria can be a problem, to say the least. Such diseases as tetanus, typhoid fever, pneumonia, syphilis, cholera, leprosy, and tuberculosis are all the nasty products of these airborne pathogens. Lucky for us, not all bacteria are this vicious; there's also 'friendly bacteria', which if it's any help to you at all is living some place private, like your pancreas, out of reach of an air purifier.
Mold spores: Mold is a fungus just like mushrooms. Mold reproduces itself through spores, which fly around in the air and aggravate your allergies. Some molds are toxic when inhaled. Black mold, which can grow on food, can produce the serious lung disease aspergillosis if large quantities are inhaled. Mold is a problem in buildings which are too air-tight and have too much trapped moisture.
Viruses: Submicroscopic and microorganisms particles that can infect the cells of a biological organism - you, for instance. Needs no introduction here. People with weakened immune systems from a present medical condition can especially do with fewer viruses in the air.
Pet dander: Dander is hair and skin particles that fly off of a pet and cause an allergic reaction in people so inclined, frequently manifested in a sneeze. Skin rashes can also be an allergic reaction to pet dander.
Dust: No, it's not a matter of how much you clean; even in the tidiest households, dust happens. Some people with dust allergies can have an extreme reaction. Another airborne dust problem is found in industrial workplace environments. If the product being manufactured produces enough dust, it can even constitute a safety hazard.
Dust mites: These little critters live in that dust that you didn't want around anyway. They're known as Dermatophagoides pteronyssinus in Europe and Dermatophagoides farinae in America, but by any name are regarded as a nuisance. Dust mites are thought to be the number one cause of asthma worldwide. Unlucky for us, dust mites like it best in the same kind of carpeted, draft-proof, cushioned environment that we people like.
Smoke: Oooooh, do we smokers ever take a lot of heat for our filthy habit! An air purifier can help you live with that human chimney in your home. It can even help the smoker's health, when they no longer have to stand outside in the snow.
Odors: The obvious one that nobody ever thinks about. Air purifiers can help dispel the taints of paint, bleach, cooking, and chemical solvents from the air. If it offends your nose, an air purifier gets rid of it.
Well, that covers most of it. It's surprising to see how much is going on right in front of our noses that we can't see, isn't it?