This may be stating the obvious for most of you guys, but just in case someone with little experience on the topic reads this thread, here's how it all works:
In order to be harmed by radiation, you need to either be in the path of some penetrating source of radiation, or you need to ingest something that is contaminated with a source of radiation.
There are several forms of radiation. Alpha and Beta radiation are pretty weak and can be stopped by thin layers of protective gear (like paper or clothing). Gamma rays and X-rays (some argue they're basically the same thing, and the line between them is blurred) are penetrating radiation, so to stop them from harming you, you need massive, dense protective layers between you and the source. For instance, you can effectively shield yourself from gamma radiation with a two-foot thick wall of concrete or a 3-foot thick wall of dirt.
The way radiation harms your body is by screwing with the DNA in your cells. With acute doses of radiation, your white blood cells will die, which eliminates your immune system, and you die from infection. This happens quickly. There are germs living in your body that are always trying to kill you, so the second your immune system is gone, you're pretty much done for. Acute radiation poisoning can cause other forms of injury, too, but the destruction of your immune system is the big one.
Now, when you receive a lesser dose of radiation, your injuries can range from nothing to irreparable harm to your DNA which leads to cancer later in life.
When a nuclear bomb is detonated, the nuclear reaction itself releases an intense burst of gamma radiation, but this burst doesn't extend much further than the fireball. Anyone in danger of being irradiated by the initial blast has bigger problems to worry about, because they just got cooked. A nuclear explosion also spews into the surrounding area the byproducts of the nuclear reaction, which include radioactive variants of certain elements, namely, radioactive iodine. Since iodine is ingested and stored in your thyroid as a normal part of being a human, the prospect of radioactive iodine getting into your body poses a real problem. If one were to ingest other radioactive products, you would most likely pass them, thus somewhat limiting the harm. It's sort of like eating an X-Ray machine and then pooping it out, so it was only able to hurt you while it was in your body. But since your body retains iodine in the the thyroid, it's a but like eating an X-Ray machine that you CAN'T poop out, so it lingers and causes longterm exposure.
The idea behind Iosat and other potassium iodide pills is that they flood your thyroid with safe iodine so that if any radioactive iodine gets into your body, it is passed rather than absorbed.
A nuclear explosion that occurs near the ground also irradiates a bunch of dirt and buildings and whatnot, which has been pulverized and spewed into the atmosphere. Said material will eventually fall back down to earth in the form of, you guess it, fallout. Each particle is a like a small X-ray machine. The only protection from these particles is shielding. They spew penetrating gamma rays, and are harmful merely by being in their vicinity. Potassium iodide pills won't protect you from that. Again, only shielding will.