Ionizing radiation is just that, ionizing. It contains enough energy to break chemical bonds. High-energy (mega electron Volt, or MeV-energy) radiation can potentially break a million or so chemical bonds, whose energy is on the scale of an electron Volt (eV). These have direct effects depending on the dose, and can break the bonds in DNA, causing mutations. These effects have obvious biological consequences, more information about which can also be found here.
The human body has natural mechanisms to repair such damage, however the best protection is prevention of exposure. For protection, all radiation workers are to adhere to the ALARA (As Low As Reasonable Achievable) principle. Following ALARA (shielding, distance, minimizing time of exposure) generally limits radiation worker exposures to about what the average person absorbs from natural sources such as the potassium in our own bodies or naturally radioactive isotopes in the soil. Radiation workers are limited to a maximum of 5 Rem/year, about 1/20th of the acute (radiation worker exposure is rarely acute) exposure required for symptoms of radiation to become obvious in the body. Most universities and state agencies require more strict limits than the federal government limit and make immediate investigation of any unusual measurement of exposure. Workers are constantly monitored with radiation badges for this purpose. Acute doses, such as would be expected in a nuclear war, are more serious than chronic doses because of the lack of time for the body to repair the damage. Chronic doses are a risk for long-term effects such as cataracts (most serious for low-energy x-ray exposure to the eyes) and cancer at higher doses.
The subject of the dose vs. the risk, particularly in the case of cancer, is a subject of intense study. Babies and fetuses appear to have the highest risk for consequences from exposure to radiation, and steps are consequently taken to protect pregnant women who would otherwise work around radioactive sources. Women also seem to have a greater risk of cancer than men if exposed to high doses of radiation. Contradictory studies of populations of radiation workers exposed to low, chronic doses of radiation over their career and nuclear weapon survivors from Japan give the counter-intuitive result that they appear to actually have lower cancer rates than the general population. Future studies based on large populations with varying levels of natural background radiation are needed to clarify this issue, but in general excessive radiation exposure is considered unsafe and avoided at nuclear facilities.