What does a reduction gearbox do in a turbofan engine?
The problem is that in bypass engines, the fan blades are much longer than the turbine blades that drive them. Both compressors and turbines should rotate as fast as possible, without shock waves occurring at the tip - so linear tip speed has an upper limit, meaning the compressor with longer blades must turn much slower than the turbine driving it.

This is what the geared turbofan does, reduces the rotational velocity relative to that of the turbine so that both can rotate at their optimum angular velocity. The forces involved and the required reliability make it not an easy design task, as noted in this answer, 


  Does it improve fuel efficiency?


It does, when both the fan and the compressor can rotate at their optimum speed, and when the gearbox does not add much friction.