There is really only one disease that we see commonly in the thyroid and that is hyperthyroidism. Cats do not get hypothyroidism unless both thyroids are surgically removed or if they have a defect in the development of the thyroid at birth, which will generally result in a very short lifespan. They can get thyroid cancer, but this is thankfully not common.
Hyperthyroidism. This is probably the second most common disease that we see in older cats. The way it works is that the thyroid develops what is called hyperplasia, not quite cancer, but kinda sorta a precursor to it, although it rarely progresses to cancer. The part of the thyroid that is hyperplastic, suddenly starts cranking out thyroid hormones into the body. While these hormones are needed in normal doses, at high doses, they increase your cats metabolism to the point that they suddenly start eating like crazy, but yet losing weight. Sounds great for us people right? Not so great. These excess hormones are also extremely hard on the organs, specifically the heart, the kidneys, the intestines, and the liver. Hyperthyroidism can cause what is called thyrotoxic hypertrophic cardiomyopathy, it can damage the kidney over time resulting in expedited chronic kidney failure, it can cause high blood pressure resulting in disorientation and caterwauling at all hours, it can predispose to inflammatory bowel disease, and it can cause thyrotoxic liver disease.
The good news, is that not only is this manageable to an extent with medication, but it can potentially be cured. Medication consists of treating with a drug called methimazole. This drug changes the thyroid hormone, making it less toxic to the body, but it still doesn't completely prevent the effects on the body organs. Most cats on methimazole live an average of 3-5 years after initial diagnosis. They usually die of either heart or kidney disease. However on methimazole they can gain weight and feel better. For a 15 year old cat newly diagnosed, that may still be a completely normal lifespan. There is also a new treatment that is dietary made by Science Diet called Y/D. This diet drastically limits the amount of iodine in the food, and without enough iodine, the thyroid cannot make excess thyroid hormone. The primary catch to this is that the diet needs to be fed exclusively, and not all cats like the taste of the food.
For younger cats, or when owners are looking for an alternative to daily medication, there are two potential cures. The first is surgical removal of the larger thyroid gland. If the thyroid hyperplasia is only in one gland, then this is a cure. If it is in both glands, we can remove both glands, but there may be a possibility that your cat would then need to be on thyroid supplementation for the rest of their life. The second is radioactive iodine. This is considered the ideal treatment. The only part of the body that take up iodine is the active part of the thyroid. With hyperthyroidism, the normal part of the thyroid becomes dormant, since the diseased part keeps cranking out thyroid hormones. Radioactive iodine is injected into your cat, and the only thing it destroys is the diseased part of the thyroid, leaving the healthy thyroid alone. This is a 96% chance of a cure. It can also be the most costly up front, costing anywhere from $700 to $2000 depending on where you have it done, and you will have to minimize your exposure to your cat for a few months afterwards. While the damage that was done to the kidneys cannot be undone, both the heart and the liver can potentially return to normal. This is the only kind of heart disease that can go away.