Phones emit non-ionising radio waves (technically in the "microwave" region of the electromagnetic spectrum, but that word is almost as likely to cause people to jump to conclusions as the word "radiation"). Cancer is caused by ionising radiation (shorter wavelengths: ultra-violet, X-ray, gamma radiation).
Although you do get people who claim it causes health effects, the only verified damage mechanism from microwave radiation is from heating, and phones are very low power transmitters. There are limits on how much radiation they can emit, or more accurately on how much the user can absorb when using them, but those are set on a precautionary basis, i.e. well below any known clinical threshold. And those are when in a call, so actively transmitting.
Epidemiologically there is no evidence that phones can cause cancer. Of course you'll find the occasional report of someone who was a heavy phone user who developed a brain tumour, but sad as that is in itself tells you nothing: you need to look at statistics, including how many people develop such tumours without phone use, and when you do that there's really no evidence. And for other health effects you have to control for other variables, some of them quite tricky such as posture (which can cause strain injuries) and nocebo effects (negative placebo: if someone thinks they are exposed to something harmful they feel or even fall ill - this is a very real thing, by the way). Again, there's no actual evidence.
There is however a small industry there to prey on people who are worried about such things, who will do anything from selling "anti-radiation sticker" placebos to put on phones to tinfoil-lining your bedroom (seriously).
