Spider mites. Spray with spinosad, you can purchase it at any big box hardware store. Follow the labels instructions and that should clear it up.

Evan

If you have a removable shower head and your tree is easily moved around, try placing the tree in your shower and washing the leavings with medium pressure cool water to get rid of the bulk of them and to clean up most of the webbing. These guys hate water and that's why they are able to take over like this indoors, because there is no wind or rain. when washing, try and wash the underside of the leaves as well because that's where they like to hide. This will only reduce their numbers to a manageable amount and clean up the webbing, you will still need to use some sort of insecticide to finish them off as recommended by Evan.

If you have any other plants in your house and especially in the same room/area you may want to examine them as well because these guys love to travel.

From the looks of things this is a pretty serious infestation and you should deal with it quickly because it wont be long before the tree starts to lose the battle and starts to decline.

Yes, and the biggest key is to keep up with them. You may kill the adults on the first action you take, but you must spray every other two to three days for at least two weeks and then once a month until the following generations are gone.