We don't know for sure about the exact relationship between the usage of mobile phones and development of brain tumours. There have been around a dozen epidemological population and hospital based studies looking into this without a firm conclusion. The fear of course is that the radiofrequency waves (RF) during mobile phone use is basically an electromagnetic radiation, although of very low frequency.
Most of the nicely conducted data is from Scandinavia (one of the largest and earliest mobile user population in the world) and a few from US; but most of the studies were deficient either in numbers, time lag and design; so no wonder a majority of the studies failed to show any clear cut risk statistically.
However, a recent study from Sweden (Mobile phone use and the risk of acoustic neuroma by Lonn S, Ahlbom A, Hall P, Feychting M; Epidemiology. 2004 Nov;15(6):653-9) has regenerated a lot of controversy again as it showed the relative risk of developing acoustic neuromas (a brain tumour of the hearing nerve; a benign tumour but tumour nevertheless) to be double in a subset of population who had been using the phones for more than 10 years. This piece of evidence does suggest that there might be some increased risk in long term users.
There is little evidence that mobile phones can cause some other tumours although symptoms of tinnitus, headaches and an increased risk of vehicular accidents are well documented.
In conclusion, while the overall evidence is still far from absolutely convincing, there does appear to be some evidence of an increased risk of acoustic neuromas, especially with long term use. This certainly needs corroboration from more scientifically well designed studies and a longer follow up. However, until we get that evidence, it may not be a bad advice to limit the use of mobile phones especially in children and young adults and phone companies should devise methods to limit the exposure of RF waves.