Transistor base voltage and current
When you're using a transistor as a switch, the typical way to turn it on is to apply a suitable current into the base pin, usually from a voltage source through a resistor. The current will then develop about 0.7 V across the base emitter diode junction. Most transistors specify a maximum base current.

In the particular case of the TIP120, this is a darlington transistor, consisting internally of two transistors. The 'base' to emitter voltage will typically be 1.4 V. 

The absolute maximum base current for the TIP120 is given as 120 mA in the data sheet. However, as the minimum gain is given as 1000, and the maximum collector current as 3 A, the base current needed to turn it fully on shouldn't need to be much more than 3 mA. This 3 mA to 120 mA window gives you a huge range to hit with your voltage source and series resistor.