'Brains' is a Stat that affects various skills requiring you to be 'smart'.
Medic, Hack, Repair, Science, Linguist, Chemistry, and Teach are all skills that rely solely on Brains, checking no other stat.
The Brains stat is also used feint in combat. When your Brains stat falls below a certain point you will get a filter which will make you sound like the retard you are.
Where to train
Read wikipedia in the matrix. Or find a certain computer deep below slagtown.