Constant current is a term most often used in electronics to describe a system that can vary the voltage across an electronic circuit to maintain a constant electric current.
When a component is indicated to be driven by a constant current, the driving circuit in essence is a current regulator and must appear to the component as a current source of suitable reliability.
An important usage of constant current power supplies are LEDs. While a high resistance series resistance is sufficient to light an LED sometimes the design needs to guard against high current. Another use is in fluorescent lamps which have very dynamic electrical resistance and are optimally operated within a short range of currents. 
^ Caladen Circuit Corner, before Sep 5 2009, LED Current Controller