Turn-on voltage for LED
A good way to think about an LED is as a variable resistance (or impedances, more commonly). At very low voltages, it has very high impedance, so not a lot of current will flow through the device. If you have a current source, as davr suggests, the current flowing through the LED times the impedance will create a voltage (Ohm's law). If you force enough current, the voltage will quickly rise and the impedance will drop. The point where current seems to flow easily is past the "turn-on" voltage you speak of. Past this point the impedance of the device is very low and large changes in current will create small changes in voltage.

I know this can still be confusing, but I would suggest staring at an I-V curve for a while (current plotted versus voltage). This is a simple looking chart but is in fact quite complex.  The thing that I did not grasp until much further into my career is that the instantaneous slope of the line is actually the impedance. Once you grasp that concept, what I said earlier may become clearer. 


(source: utc.edu)