What is the bypass air in a turbofan engine actually for?
On modern turbojets the bypass air provides (at least) two things 


Thrust

The air does bypass the engine core, but it is accelerated by the N1 fan and provides thrust as it is expelled rearward out of the engine.  Thrust from the bypass air can contribute more than half of the total thrust produced by the engine (upwards of 80% of the total thrust for some engines in certain phases of flight).  Part of the reason the bypass air is so efficient is that much energy is extracted from the core exhaust to spin the high and low pressure turbines (dual spool) that drive the N1 fan and the N2 compressor stages.
Engine cooling and noise reduction

Cool bypass air can be mixed with the hot air that went through the core at the rear of the engine.  This mixing cools the engine exhaust and reduces the jet engine noise resulting from the rapid expansion of that air once it leaves the engine.  The cool bypass air flowing around the engine core can also be used for general engine cooling.