Electromagnetic radiation can mean many things. Short wavelength UV, X-Rays, and Gamma rays are all ionizing radiation. Ionizing radiation is energetic enough to dislodge electrons and alter molecular structure. It is widely known that ionizing radiation is harmful to living tissue. Very powerful non-ionizing radiation is known to be able to cause damage from tissue heating (e.g. microwave ovens). The danger here is determined by the frequency and strength of the radiation. Microwave frequencies in the GHz range are the most readily absorbed by tissue so they are the ones most likely to damage tissue through heating. Frequencies above and below this range mostly pass through tissue harmlessly without being absorbed. The damage Ionizing radiation causes is cumulative. X-Rays cause some tissue damage but when doctors need diagnostic images the benefits outweigh the risk. Someone being unnecessarily X-Rayed multiple times could be injured. In theory non-ionizing radiation too weak to cause tissue warming should be harmless. Currently the effects of very weak ionizing radiation and of non-ionizing radiation too weak th heat tissue are being studied. Some believe that this radiation can cause damage through some mechanism not yet understood but so far the results are inconsistent and inconclusive. For example, radiation from living under high tension power lines should be harmless because the low frequency but some researchers suspect a cancer link.
First of all electromagnetic radiation carries energy. That energy can turn into heat and damage cells. Some old mobile phones were suspected to cause cancer that way - as several users of these cells had brain tumors right above the ear, next to where the antenna ends.
Microwaves also work with electromagnetic fields. Water molecules have a positive-pole and a negative-pole. The microwaves "shake" the water molecules. Matter with fast moving - energy-rich molecules is hot. I don't have to tell you what happens to a person who puts their head into a microwave.
The other thing is that the signals in nerve cells are transported by electromagnetic charges. So a electromagnetic wave in the right frequency might interfere with nerve signals. Some people who are sensitive to this claim that they get headaches and concentration problems from electromagnetic impulses. Mobile phone antennae and radio senders are often complained about. Some scientists doubt this though... They think that these symptoms are psycho-somatic (means they only exist because people believe in it).