Mobile Phones work using radio waves, which are non-ionizing in nature unlike X-rays or gamma rays. It can heat up tissues if the radiation intensity is high but wont ionize tissues.
Lot of researches are done across the world to demystify the question of the relationship of cancer with the use of mobile phones. But no significant results are found out yet to prove the same. The Danish Study in 2001-2006 claimed that there is no link between cancer and mobile phones, but a Swedish research team in 2009 found out cancer chances in people who use mobile phones for prolonged period, though the exact cause/mechanism was not found out.
A term which deserves attention here is Specific Absorption rate (SAR) — the amount of radio frequency energy absorbed by the body when using a phone. The value should be minimum for a good phone which cause no/minimum health problems. The United States Federal Communication Commission (FCC) standard for SAR limit of 1.6 watts per kilogram averaged over a volume of 1 gram of tissue for the head.
Make sure your Mobile phone SAR value is less than this limit. Limit the time you spend talking in the phone and switch to other modes of data transfer like texting. Users in age limit below 20 years should take particular care not to use mobile phones for long and use devices like hands-free if you need to make long calls.