I hope this makes some sense:
http://www.elperfecto.com/2011/01/04/calculate-resistor-values-for-driving-leds/
Usually, you don't drive from the emitter, but the author of your link had reasons to do so (miller capacitance:
http://en.wikipedia.org/wiki/Miller_effect). This would make the equations different.
Basic equations for a transistor are: Ie=Ib+Ic
There is a voltage drop depending on temperature from base to emitter: Vbe which is around 0.6V for silicon.
There is a saturation voltage from Collector to Emitter of Vce(sat). The voltage won't be lower than that.
Transistors have a current gain called hfe, It also has another gain when used when saturated. Hfe has a huge variation. Both per transistor type and overall. Hfe from about 20 to 20,000. The 20,000 is for a darlington transistor configuration.
Ic=Ib*Hfe
When designing an amplifier, the idea is to make the circuit work for various values of Hfe. This is really not an amplifier,. It's a switch.
LEDs have a voltage drop depending primarily on color. This is on the order of 1.2 ro 3V