First, it is useful to sort out radiation into two categories - non-ionizing and ionizing radiation. Only the latter is of serious concern to long-term health effects. Electromagnetic radiation at visible energies and below generally lacks sufficient energy to cause disruptions to cells or DNA. At ultraviolet energies and above, radiation can cause cellular damage (as Cyrus pointed out, this is exactly what is happening when you get a sunburn...)
But even this picture isn't quite complete; namely, because not all radiation is created equal. While most of what we consider "radiation" (particularly, ionizing radiation) falls into the categories of X-rays and gamma rays, there's more to it than this.
First, there are three different types of ionizing radiation, including ionizing radiation from photons, alpha radiation, and beta radiation. Each of these different types of radiation comes from a different source and can be stopped in different ways.
Electromagnetic radiation: photons
X-rays and gammas are photons like radio waves and visible light, only having shorter wavelengths and higher energies. X-rays are emitted when electrons change energy state in the atom, while gammas result from changes of state in the nucleus itself (including fissions). How deeply penetrating EM radiation is depends upon its energy; the more energetic the photon, the thicker the shielding required. Generally speaking, photons require the greatest amount of shielding, being "deep" penetrating radiation. This type of radiation is best stopped by materials with high atomic number, with dense materials such as lead.
Interestingly enough, the fact that photons pass through less dense materials (and are stopped/absorbed by denser materials) is how an X-ray scan works. Bone is much more dense than blood vessels or soft tissue - thus, X-rays pass through with much greater frequency than they do bone. By exposing a film behind the object, an X-ray is used to scan internal objects (like bones) by looking for the spots on the film where fewer X-rays arrive (i.e., they've been stopped by bone).
|(Image source: Wikipedia)|
"Alpha" particles are really just a fancy term for a helium nucleus with no electrons. Compared to other forms of radiation, it's big, heavy... and slow. Alpha radiation is the least penetrating type of radiation, capable of being stopped by a thin piece of paper or the outermost layers of your skin. Alpha radiation is thus not generally dangerous unless ingested into the body, either by eating or breathing.
Alpha radiation tends to be special in that it is generally ejected from very heavy nuclei like uranium, thorium, and other very heavy elements, along with their decay products. For example, if you live in an area with a large amount of natural granite deposits, particularly if you have a basement, chances are good that you've had to have your basement tested for radon gas. Radon is a decay product of natural uranium in the ground (which is often found along granite deposits).
|The uranium decay series (courtesy: Wikipedia)|
While radon is a noble gas (and thus of little concern on its own), the health hazard comes from ingesting radon into the lungs, where it decays into polonium (which sticks inside the lungs, producing several "daughter" alpha decays).
Likewise, if you smoke, chances are you're getting a heavy dose of radium directly to your lungs, as radium tends to "stick" to tobacco leaves, where it is ingested when the tobacco is burned and inhaled; this radium decays via series of rapidly decays to Polonium-224 - the same thing used to poison former KGB agent Alexander Litvinenko a few years back. (Another very good reason not to smoke!)
So-called "beta" radiation consists of electrons and their anti-particle, "positrons." Beta radiation is produced when a nucleus decides to "change" its configuration. Nuclei tend to prefer a certain ratio of protons to neutrons - too many protons, and the nucleus will try and "push" itself apart from the repulsion generated by like charges. Neutrons tend to act as a "buffer" in the nucleus, along with the strong force which binds the nucleus together.
However, despite being electrically neutral, nuclei tend to prefer not having "too many" neutrons either. When the number of protons or neutrons is "out of balance," nature will often employ the weak force to change the "flavor" of a proton (or neutron) to a neutron (or proton). In this process, electrical charge is conserved - a (negatively charged) electron or a (positively charged) positron are emitted. Likewise, a "ghost" particle known as a neutrino is also emitted - "ghostly" in that they have very small (almost non-existent) mass and no charge (and thus they barely interact with matter).
|Beta radiation from a nucleus and free neutron, respectively. (Courtesy: Wikipedia)|
Beta radiation can generally be stopped by a thin piece of metal. While it is more penetrating that alpha radiation (beta is considered "shallow" penetrating radiation), for an external exposure it is generally a minor concern except for sensitive areas such as the lens of the eye; the chief concern for beta radiation again lies with ingestion.
Radioisotopes such as iodine and cesium fall into the category of beta emitters.
The dose makes the poison
When Cyrus mentioned the issue of radiation being (mis)-characterized in terms of its raw activity (i.e., the counts per minute), measurements of counts per second tell us nothing about the type or energy of that radiation. A raw count of radiation doesn't tell us what type of radiation being received (gammas, betas, or alphas), or what energy this is at. The difference is quite extreme - while no one would want to be contaminated with an alpha source, the solution would be quite simple - a change of clothing and vigorous scrubbing.
Second, doses are typically measured not by radiation counts alone, but by folding together both the count of particles (by type) with their energy. Radiation workers tend to be carefully tracked for their exposure to radiation through use of "TLD badges" (which stands for thermoluminescent dosimeter). This is where units such as rem and Sieverts tend to come into play - these units measure both the energy absorbed by the body as well as making a relative adjustment for the radiation type (called a "quality factor," which accounts for the fact that different types of radiation have different impacts). Thus, any relevant dose numbers tend to be reported in these units, rather than raw counts alone.
If you've ever been to a facility which makes regular use of radiation (such as a hospital or even possibly your dentist), you may have seen a badge looking like this:
|A TLD ring and badge (Image courtesy of Princeton University)|
Radiation effects are generally broken down into two types of categories: acute and stochastic (i.e., "random"). Acute effects generally only occur at very high exposures - these symptoms are caused when there is significant amount of damage to internal cells in the body (such as gastrointestinal cells and the nervous system). Symptoms of acute radiation exposure tend to include skin redness and irritation (erythema), nausea and vomiting.
Stochastic effects are produced from long-term exposure to lower levels of radiation, taking years or even decades to produce effects. Typically, this is what is thought of when people think of radiation's carcinogenic (cancer-causing) effects. Our estimates of stochastic radiation effects tend to be extremely limited, largely based upon studies of atomic bomb survivors in Japan. Generally speaking, the current assumption tends to extrapolate from known radiological exposures of atomic bomb survivors, who received relatively high doses, drawing a straight line backwards to zero dose. This is called the "linear no-threshold" (LNT) model of radiation, which assumes all radiation dose has some intrinsic probability of causing a cancer.
Experts disagree about the applicability of LNT model, due to the extremely limited data for lower radiological exposures - particularly given the fact that humans are regularly exposed to ionizing radiation from natural, background sources as well (such as from natural radiation sources in the ground, as well as others such as in potassium-40, found in bananas and other foods such as Brazil nuts). However, models such as the LNT model tend to form the basis of current exposure guidelines, such as what a normal acceptable dose for members of the public and for trained radiological workers can be, based on these assumptions of cancer risk per unit dose.
NEI: Radiation in perspective
This guide (PDF) from the CDC gives a basic overview of units of measure of radiation, including commonly used prefixes (such as pico, micro, milli, etc.)
CDC: Radiation emergencies FAQ including a brief explanation of radiation health effects and how exposure can occur.