That is a bad case of spider mites. Check the link to a similar question. The good news is that you can control the situation easily and safely with soap and water and a cloth.
Mix up a solution of a teaspoon of dish soap to a quart of water and stir. Soak the cloth in the solution and wipe down every surface of the plant, top and bottom, stems, everything.
Wait five minutes and then rinse or spray the plant with water to wash off the soap.
The Spider mite eggs that are left will hatch in approximately five to seven days at room temperature. You will have to repeat the treatment at least twice and continue watching for them to return.
Edit: I reviewed the picture and it's possible that you have mealy bug as well. There are some white cottony tufts in the upper left of the first photo. Soap and water will help control them too.
The damage to the leaves from the spider mites is permanent. If the plant was in good health you could consider waiting until things are under control and then pruning it back and removing the damaged foliage. This should cause new growth which will be healthy.
Edit: Sandra asks where do spider mites come from? For new plant purchases it is most likely that spider mites were present in the grower's greenhouse or holding area but were being controlled by pesticide sprays. After you bought the plant the population started increasing. For an established plant spider mites can transfer from other plants in the immediate vicinity.
Citrus plants are known to be tasty to spider mites and the low humidity conditions found indoors are a low stress condition for most houseplants.
Another solution, if you live in a warmer area, is to put the plant outside, under shade and hose it down. This will provide a rapid take down of the population and make it easier for subsequent applications of soap and water to control the population.