I have a box of salt in the kitchen, and it says, "Iodized Salt*", and then at the bottom it says, "*This salt supplies iodine, a necessary nutrient". On the ingredient label is lists Potassium Iodine at a concentration of 0.006%. A quarter teaspoon of salt (1.5 grams) provides 67 micrograms of iodine, which is about half of the U.S. Recommended Daily Allowance for iodine.
The main reason that you need iodine is because of a gland in your neck called the thyroid gland. The thyroid gland produces two hormones (thyroxine and triiodothyronine) that your body uses during metabolism (see this article for details). Without these hormones you start to feel tired, depressed, cold, weak, etc. Iodine is an important element in these two hormones, so without iodine your thyroid gland cannot produce them. When starved for iodine, the thyroid gland also swells, and when it does it is called goiter (see this page for a picture).
Your body doesn't need or contain very much iodine. You might have 20 to 25 milligrams of iodine in your entire body right now. However, in some parts of the world the soil contains no iodine, so the plants contain no iodine and therefore iodine deficiency is a problem. In the U.S., one part of the country that lacks iodine is the Great Lakes region. So companies started adding iodine to salt in the 1920's to eliminate goiter and thyroid problems.
If you lived through the cold war, you may have heard about the practice of taking iodine pills during the threat of a nuclear attack. When a nuclear bomb explodes, one substance it forms is radioactive iodine. If you eat, drink or inhale this isotope, your thyroid gland will concentrate it and this can lead to thyroid damage or cancer. By taking an iodine pill, you saturate the thyroid gland with iodine and prevent it from absorbing the radioactive iodine.
Here are several useful links: