Microwave is something of a catchall term for any form of electromagnetic radiation from the middle of the UHF range (300-3000MHz) up to 300GHz or so. In terms of wavelength, this is a range of 1 millimetre to 1 metre, which can be shown by a cool trick with some marshmallows and a microwave oven. The term was probably coined by analogy with "shortwave" and similar terms used in radio broadcasting, based on the wavelength of the radiation.
Microwaves are widely used for radio transmissions, mostly for line-of-sight or quasi line-of-sight applications. WiFi gear and cellular phones, for example, operate on the lower ends of the microwave range. Microwave bands are widely used for line-of-sight high bandwidth links, for instance to carry telephone calls between switchboards. Microwave radiation is also used in radar, heating (microwave ovens are used in both cooking and in some industrial and medical applications), and the like.
 Cancer fears
For the last thirty years, since the publication in the late 1970s of a series of articles by investigative journalist Paul Brodeur, microwaves have been considered a cancer risk by a great many members of the public. Widespread fear of the admittedly rather loaded word "radiation" has led to considerable panic over such things, starting with microwave ovens and radar installations, proceeding to worries over power lines (the radio emissions from which are actually in the super low frequency band at 50 or 60 Hz), CRTs, mobile phones, and even electronic "smog" from WiFi transmitters. Cancer fears have led to reduced property values near power lines, protests of radar installations, and even attempts to banish wireless networking from schools.
Further worries have grown from fears that microwave heating does something "different" or "unnatural" to foods being processed, and are thus harmful. In the manner of all good health scares, this has been aimed at children, with a lot of attention on reheating breast milk. However, there is little evidence to suggest that it is microwaves per se that cause the issue. Any protein denaturing effects are mostly likely due to localised hotspots forming within the microwave oven - as the marshmallow experiment mentioned above will show - which raise the temperature well above body temperature. This same effect can be replicated with any heating method if it is applied to directly or with insufficient mixing of the liquid.
Epidemiological studies have not borne out the feared risks, however; in fact, it's generally thought that the fear springs from a lack of knowledge about the difference between ionizing and non-ionizing radiation. While microwaves can have chemical effects on the human body, they're the sort of effects indirectly caused by heating, such as protein denaturation, burns from heated water in tissues, and electric arcing from metal objects in the microwave field. These risks are not negligible with industrial microwave ovens. Ionizing radiation on the other hand, which begins around violet in the visible light portion of the electromagnetic spectrum, is where the cancer risk lies — the radiation in question is energetic enough to directly break molecular bonds. Microwaves do not have that level of energy; needless to say, the far lower frequencies of power lines do not either.
 WiFi "smog"
IEEE 802.11, aka WiFi, is a group of standards designed for wireless networking of computers in the microwave radio spectrum, primarily in the 2.4 and 5.8 GHz bands. It is based on work going back to 1970, when the very first wireless networking system (ALOHAnet) was implemented at the University of Hawaii by a surfer dude and ham radio operator named Norman (who also happened to be a computer science professor, which might have had something to do with it). WiFi, particularly in its 802.11n and 802.11g variants, is the dominant wireless LAN protocol on the market.
We mention WiFi here because in some quarters, it has been lumped in with mobile phones and power lines in the microwave hysteria; in fact, school and municipal WiFi projects have occasionally been put on hold because of fears of "WiFi smog". In actual practice, WiFi gear usually puts out an absurdly small signal, generally no more than 200 mW, and usually has a range of no more than a hundred meters (if that). What is a real concern is that WiFi (at least b, g, and single-band n) operate within the very crowded 2.4 GHz radio band, leading to colossal amounts of radio traffic that tends to cause particular problems with cordless phones.
Radiation in the 2.4 GHz band is readily absorbed by water and is thus very good at heating it. This is the reason microwave ovens are designed for this frequency range. As a result, the band is full of interference from the ovens, and it is an officially-designated "garbage" band where you can pump out any crap you like up to a few hundred milliwatts without a license. Which is why WiFi uses this range, and which is why Wifi-a and the upper band of n live in the 5.8 band and DECT cordless phones live in dedicated bands on 1.9 or 1.8 GHz (depending on the country).
WiFi fear is going strong in 2012.
 Further reading
- Park, Robert, Voodoo Science, 2000. Covers the issue of microwave hysteria, starting from Brodeur's ass-backwards investigation. Doesn't include cell phones, as the cell phone hysteria was just about hitting its peak when the book was written.
- Federal Communications Commission - Questions and Answers about Biological Effects and Potential Hazards of Radiofrequency Electromagnetic Fields
- The Claim: Microwave Ovens Kill Nutrients in Food, New York Times
- If your house lacks babies, this may be why.