They are spider mites which are a very common houseplant insect. The chemical control that can be used is called Safer Soap and can be purchased anywhere with a garden center. You can also use an organic horticultural oil such as Neem Oil. However, if you have children or pets and do not want to use an insecticide, you can also bring your plant outside and spray it fairly hard with your garden hose which will knock the mites off.
By Nancy (Guest Post)
11/02/2007
These are called spider mites. There is a spray that you can get - I can't remember it (too long since I worked in greenhouse) - and you'll want to put it away from other plants as it will 'spread' to your others. If too far gone the plant will die. If I remember right, the spider sort of drills into the main stem and it can die from this. Good luck.
By
10/26/2007
I forgot what they're called but you can get rid of them with rubbing alcohol. Dip a Q-Tip in alcohol and clean the web off. It won't hurt the plant.