Electromagnetism refers to the radiation of tiny oscillating electric and magnetic fields, commonly called photons. Just like in the old high school experiment, where the flow of electric current through a coil of wire induces a magnetic field and the movement of a magnet inside a coil of wire induces an electric current, any movement of an electric charge through space induces a magnetic field. Similarly, a moving magnetic field induces an electric charge, a principle that forms the basis of almost all modern electricity generation. Thus a moving photon of electrical energy will induce a small moving magnetic field, which will in turn induce a small moving electric charge, and so on. The term electromagnetic is used to describe this backwards and forwards oscillation between electric to magnetic energy.
The speed at which these oscillations occur depends on the amount of energy being radiated by the source. A graphical representation of radiometric energy at a range of frequencies is called a spectrum. Different forms of electromagnetic radiant energy can be laid out to form a continuum from very low frequency radio waves all the way through to very high frequency cosmic rays. Taken as wavelengths, the electromagnetic spectrum ranges from around 1e5 metres (100km) for low frequency electric current to 1e-15 metres (0.000000000001 mm) for cosmic rays.
Radio and Radar Waves
These wavelengths represent the lowest end of the electromagnetic spectrum, though some would argue that alternating electrical current (AC) falls below. Radio and radar waves pass straight through the atmosphere relatively unimpeded and surround us constantly. Very high levels of exposure to this form of radiation has been linked to cancers in some animals (mainly laboratory mice for some reason). There was a story going around that, on very cold winter mornings during World War Two, British soldiers would gather in front of the large radar dishes at the major communications stations. As the upper end of this spectral band is often referred to as microwaves, there was a real reason why they felt a little warmer standing there.
Long Infra-red
Above radar waves is the long infra-red band, often called terrestrial radiation. These long, low energy wavelengths are the radiation emitted by terrestrial objects at terrestrial temperatures - between -10 and 100°C. Biological metabolisms also generate heat which radiates from the skin surface at these wavelengths.
Short-wave or Near Infra-red
Just above this is the near infra-red band. This is the range of frequencies that give the physical sensation of heat. It is termed near infra-red as it is quite close to the visible band, and causes the sensation of heating we feel on our skin when directly exposed to the Sun or an open flame. See the greenhouse effect topic for information on how the differences between short and long-wave radiation can be exploited to effectively heat buildings.
Visible Light
The black segment of the spectrum above represents visible light. Within this of course are all of the wavelengths that constitute the colours of the rainbow, from red at the bottom to violet at the top. If you are interested, to recall the major components of the visible spectrum (working from the lowest frequency to the highest) try to remember that guy called ROY G. BIV - (Red, Orange, Yellow, Green, Blue, Indigo and Violet).
NOTE: It used to be common to call the region between blue and violet "indigo". In modern usage however, indigo is not usually distinguished as a separate colour; thus you may sometimes see Roy without any vowels in his last name; hence "ROY G. BV".
Ultraviolet
Whilst overexposure to ultraviolet light has been linked to skin cancer, some exposure is necessary for the bodily production of certain vitamins, and to the healthy growth of plants. It is primarily UV radiation from the Sun that fades fabric and degrades many building material finishes.
X and Gamma Rays
Exposure to very high frequency X and gamma rays is dangerous to human beings, causing radiation sickness and cancers. Fortunately, gasses in the upper atmosphere absorb almost all of these frequencies from solar radiation before they reach the Earth's surface. It's the leakage from nuclear waste sites you really need to look out for.