The main danger from radioactivity is the damage it does to the cells in your body.

Most of this damage is due to ionisation when the radiation passes, although if levels of radiation are high there can be damage due to heating effects as your body absorbs the energy from the radiation, rather like heating food in a microwave oven. This is particularly true of gamma rays.

Alpha Particles (a)

Alpha particles are slow, have a short range in air, and can be stopped by a sheet of paper.
You might therefore assume that alpha particles are the least dangerous of the three types of radiation.

Wrong! Whilst they cannot penetrate your skin, you could easily eat or drink something contaminated with an a source. This would put a source of a particles inside your body, wreaking havoc by ionising atoms in nearby cells. If this happens to part of the DNA in one of your cells, then that cell's instructions about how to live and grow have been scrambled. The cell is then likely to do something very different to what it's supposed to do, for example, it may turn cancerous and start multiplying uncontrollably.

Thus alpha particles, whilst they have a low penetrating power, can be the most dangerous because they ionise so strongly.

b-particles have a longer range than a's, but ionise much less strongly, with the result that they do around 1/20th of the damage done by the same dose of alpha particles.

However, they do have more penetrating power, which means that they can get through your skin and affect cells inside you.

Click to find out about units

Gamma rays hardly ionise atoms at all, so they do not cause damage directly in this way.
However, gamma rays are very difficult to stop, you require lead or concrete shielding to keep you safe from them. When they are absorbed by an atom, that atom gains quite a bit of energy, and may then emit other particles. If that atom is in one of your cells, this is not good!

You can read more about gamma rays in the Electromagnetic Spectrum web site.