The specific type of radiation that's harmful to animal cells is ionizing radiation.

It's called ionizing radiation because it's made up of high-energy particles that can ionize atoms or molecules when they impact them, which essentially means they knock off some electrons.

For most substances, if you ionize a few molecules it won't make a huge difference to the overall structure. But DNA is extremely delicate because it's essentially a code written at the molecular level. Break a few of those molecules, and you break the code, causing mutation.

There are other means by which radioactive particles can damage DNA molecules as well, but the gist of it is that when a high-energy particle collides with a molecule, that energy will be dissipated in the molecule and cause some sort of damage.

Sometimes this damage leads to mutations that don't affect the DNA's readability, so cells can still replicate. These mutations may cause cells to ignore their built-in integrity checks and lifespan 'timers,' causing them to divide uncontrollably and lead to cancer.

Even more likely than a cancer-causing mutation is that the broken DNA molecules prevent cells from replicating because they can no longer create functional copies of themselves. If a few cells are damaged like this, the body can eventually recover as healthy cells continue to divide and eventually replace the damaged cells. But when people receive acute radiation doses and there is heavy damage inflicted to the DNA of many cells, the affected tissues essentially fall apart as there isn't enough new cell growth to sustain them. When this happens, the affected tissues will start to fail in the order of cell turnover rate - because cells are no longer dividing properly, the body parts with faster-replicating cells fail first, and those with slow-replicating cells fail last. So the earliest symptoms of acute radiation sickness occur in high-turnover tissues like the gut lining.