Iodine is absolutely an essential mineral, and is required for proper functioning of your body. It is a component of the thyroid hormone triiodothyronine (also known as T3) and its precursor form T4 (thyroxine). This hormone is involved in quite a few processes in the body, and acts on nearly every cell in it.
A deficit of iodine can cause a number of diseases, including goiter (swelling of the thyroid gland in the neck), hypothyroidism (can cause increased sensitivity to cold, fatigue, weight gain, constipation, and depression), gastric cancer, and cretinism (presents as severely stunted development and growth, both physical and mental, but unlikely to affect a healthy adult).
Now, iodized salt is not the only source of iodine in one's food. It can also be found in fish, shellfish, and kelp products, milk and eggs from iodine-sufficient farm animals, and plants grown on iodine-rish soil. The recommended intake of iodine varies between 150 µg per day for healthy adults to 250-290 µg/day for pregnant and lactating women, respectively. Children require somewhat lower amounts. On the other hand, the recommended upper limit is around 1100 µg (1.1 mg) per day for adults.
A quarter teaspoon (1.5g) of the iodized sea salt in my cabinet provides "45%" of the recommended daily value of iodine (the label doesn't provide mass). Assuming they're using the 150 µg value, that's 67.5 µg iodine in a serving that provides one quarter of your sodium for the day. The table in this section of the Health Professional Fact Sheet for iodine from the NIH's Office of Dietary Supplements provides some values for iodine levels in various other food sources:
So, bottom line, you are likely getting sufficient iodine in your daily diet, even without using iodized salt. Now, I don't want to discourage the use of it at all, as there is absolutely nothing wrong with it from a health perspective (except from overconsumption, but that applies to most everything). Not everyone (especially in America, the UK, and some other countries) eat very healthy diets, but since the use of iodized salt in restaurants and for food manufacturers is required by law, Westerners are not likely to be under-iodinated.
Of course, the best way to tell is to go to your primary health care provider, explain your situation, and ask if they'd be willing to order a thyroid function blood test. It looks at the levels of thyroid stimulating hormone (TSH, from the pituitary gland) along with thyroid hormone itself. However, as long as you aren't experiencing any of the symptoms of hypothyroidism and don't have a goiter, you're probably OK.