Cat dander is dead skin cells. Cats, and most other warm blooded animals including humans, shed their skin in tiny flakes in order to constantly replace old skin with new healthy skin. This is performed in some way by almost all living creatures so to help them stay healthy by warding off disease and infection. The constant replacing of skin cells is how creatures heal themselves. The 'dust' that is on the surfaces in your house is actually dander. Ever member of your family, and your pets leave a small amount behind every day.
Most people believe that dander is invisible but it is actually very very small. While you can see the dust left on the surfaces in your house, it will require a microscope to see pieces of individual dander. Even if you cannot see the clouds of it traversing your house or coming out of your couch everytime you touch its surfaces doesn't mean it isn't there. It is also a common belief that if you do not have a cat you will not be effected by cat dander. This has proven to be false. Even office buildings, which have never had cats in them, have traces of cat dander in them. This is the result of people with cats at home bringing small amounts of dander with them to work on their clothing. Luckily, you will have to have an extremely severe allergy to cats for this to effect you.
A lot of people also think that they can eliminate all of the dander from their living area simply by vacuuming and dusting. It is simply not possible to completely remove dander from your house once you have had a cat in it. However, it is true that vacuuming and dusting your house often can cut down your allergic reaction to cats incredibly.