Radiation and "Safe Dose"
Great question. It is unfortunate that more people don't understand what &quot;radiation&quot; is and how it affects biology.
Firstly, you must distinguish between ionizing radiation and non-ionizing radiation.
Wikipedia describes the former rather well:

Ionizing (or ionising) radiation is radiation composed of particles that individually carry enough kinetic energy to liberate an electron from an atom or molecule, ionizing it. ...
When ionizing radiation is emitted by or absorbed by an atom, it can liberate an atomic particle (typically an electron, proton, or neutron, but sometimes an entire nucleus) from the atom. Such an event can alter chemical bonds and produce ions, usually in ion-pairs, that are especially chemically reactive.

X-rays, alpha, beta, and gamma radiation are all ionizing forms of radiation. Non-ionizing radiation, as you might guess, is radiation of a sort which lacks enough energy to to strip an electron from an atom. All light is actually &quot;radiation.&quot;
The reason why ionizing radiation can be so damaging is because the molecular underpinnings of our cells depend on many very specific chemical bonds. Changing those bonds can damage or even break anything from proteins to strands of DNA itself. Our bodies naturally try to repair damage, but the amount of damage is a crucial factor in how successful our bodies can be. More radiation means more damage; also greater amounts of radiation increases the chances of serious genetic damage to one or more cells. Damaging a cell's rulebook in just the right ways can disable its repair processes and even its fail-safes that keep it from becoming a hazard to the rest of the body (such as growing uncontrollably into a cancerous tumor).
A more in depth explanation of radiation effects can be found at http://www.epa.gov/radiation/understand/health_effects.html#q1

In general, the amount and duration of radiation exposure affects the severity or type of health effect. There are two broad categories of health effects: stochastic and non-stochastic.
Stochastic Health Effects
Stochastic effects are associated with long-term, low-level (chronic) exposure to radiation. (&quot;Stochastic&quot; refers to the likelihood that something will happen.) Increased levels of exposure make these health effects more likely to occur, but do not influence the type or severity of the effect.
Cancer is considered by most people the primary health effect from radiation exposure. Simply put, cancer is the uncontrolled growth of cells. Ordinarily, natural processes control the rate at which cells grow and replace themselves. They also control the body's processes for repairing or replacing damaged tissue. Damage occurring at the cellular or molecular level, can disrupt the control processes, permitting the uncontrolled growth of cells cancer This is why ionizing radiation's ability to break chemical bonds in atoms and molecules makes it such a potent carcinogen.
Other stochastic effects also occur. Radiation can cause changes in DNA, the &quot;blueprints&quot; that ensure cell repair and replacement produces a perfect copy of the original cell. Changes in DNA are called mutations.
Sometimes the body fails to repair these mutations or even creates mutations during repair. The mutations can be teratogenic or genetic. Teratogenic mutations are caused by exposure of the fetus in the uterus and affect only the individual who was exposed. Genetic mutations are passed on to offspring.
Non-Stochastic Health Effects
Non-stochastic effects appear in cases of exposure to high levels of radiation, and become more severe as the exposure increases. Short-term, high-level exposure is referred to as 'acute' exposure.
Many non-cancerous health effects of radiation are non-stochastic. Unlike cancer, health effects from 'acute' exposure to radiation usually appear quickly. Acute health effects include burns and radiation sickness. Radiation sickness is also called 'radiation poisoning.' It can cause premature aging or even death. If the dose is fatal, death usually occurs within two months. The symptoms of radiation sickness include: nausea, weakness, hair loss, skin burns or diminished organ function.

Another thing to understand about why the dose matters is that we are surrounded by ionizing radiation. We always have been. Our bodies have evolved needing to successfully deal with ionizing radiation. You can find a very informative and fun visualization of this fact at https://xkcd.com/radiation/.