The FCC has yet to implement GAO’s recommendations to more closely reflect real-life use. For a narrow subset of smartphones – those sold with lanyards or straps – the FCC advises manufacturers to test phones at a distance of no more than 5 mm from the body (FCC 2014). Yet the FCC has done nothing to ensure more realistic testing of most other smartphones or to account for the widespread use of accessories such as cases, which many different manufacturers produce with both metallic and non-metallic components.
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The exact source of radiation in a cell phone is from the transmitter, a device located near the antenna that converts audio data into electromagnetic waves. The amount of radiation a cell phone can emit is limited by legal restrictions in the U.S., Canada and Europe. Additionally, the average radiation levels of most mobile phones are available to the public, courtesy of the Federal Communications Commission in the U.S.
In this frequency range, the interaction between matter and light is via the electric field component of light (totally different from how ionizing radiation messes you up). In particular, an oscillating electric field causes polar molecules to rotate or attempt to rotate, and the lag between the applied field and the response of the molecules manifests as dissipation—i.e. heating . This is the same dielectric heating which is the operating principle behind microwave ovens (which operate at 2450 MHz  similar to wifi). So if cell phones were to cause damage to tissue, the mechanism would be the same as what happens in a microwave oven—boiling the water in your head/body.
The energy of electromagnetic radiation is determined by its frequency; ionizing radiation is high frequency, and therefore high energy, whereas non-ionizing radiation is low frequency, and therefore low energy. The NCI fact sheet Electromagnetic Fields and Cancer lists sources of radiofrequency radiation. More information about ionizing radiation can be found on the Radiation page.