All kinds of radioactive isotopes might enter or attach to your body in many ways as consequence of a nuclear accident. Any radioactive particle that gets close to or into your body will cause some damage depending on the type (alpha, beta, gamma), strength of radiation and duration of exposure.

If you are not exposed to an immediately lethal dose, the damage in the body is a statistical thing. The longer and the more intense the radioactive particles remain on or in your body, the more likely is the chance for a serious health issue. The strategy is to remove radioactive particles as fast as possible to minimize the probability of damage.

Radioactive particles on your skin can be washed away. (This is not so fun as the every-day shower).

Radioactive particles that you inhaled are a really bad thing, I do not know how to deal with radioactive particles in the lungs (except wearing filter masks in time).

Particles that entered the body with food will be excreted within a couple of hours.

But: your body has an organ - the thyroid gland - which collects iodine from your food and stores it long time. A radioactive form of iodine - iodine 131 - is a common byproduct of nuclear contamination. If this gets stored in your body, it remains long enough to dramatically raise the chance of cancer. Therefore you fill your thyroid gland with non-radioactive iodine, hopefully BEFORE your body gets contaminated with radioactive iodine.

So the special thing with iodine 131 is, that it is common, that the body does not make a difference to nonradioactive iodine and there is an organ that stores it.

By taking iodine pills, you only improve probabilities on that one factor! There are plenty other ways to take damage from radioactive accidents.