As snailplane said, you can easily answer the question by thinking that there is normally an article in front of a singular count(able) noun, which is brain in this case. A singular countable noun almost always requires an article, which could be definite or indefinite, depending on what the speaker or the writer means.
The following could answer your question about why the is used here (taken from Practical English Usage by Michael Swan, page 65, entry 70.15):
When talking about parts of someone's body, or about their possessions, we usually possessives, not the
But the is common after prepositions, especially when we are talking about blows, pains and other things that often happen to parts of people's bodies.
She hit him in the stomach.
(Non-italic, bold emphasis mine)
In the sentence "Many people are worried about the bad effects of mobiles on the brain", the bad effects of mobiles is what happens to the brain; it's also after a preposition (on). So the fits here.
Please note that there are a lot of the uses other than its main use, although most of them probably could be explained saying how the uses are somehow specific/definite. The above is only one of them; in the same book from which I took the excerpt there are 19 sub-entries for special rules and exceptions for the, and I doubt that's all there is.