BJT transistor as an amplifier
A short answer: The bipolar transistor (BJT) works as a voltage-to-current converter for (small) input voltage variations around a suitable DC bias point. In any case (for all three basic configurations), this input voltage is the voltage (change) between the base node and the emitter node. 

The corresponding voltage-to-current relationship can be verified by the exponential transfer characteristic Ic=f(Vbe), which first was described by W. Shockley. The parameter which describes this relationship is the transconductance gm=d(Ic)/d(Vbe).