In
simple terms:
Darlington driver = a high gain pair of transistors where you don't need much current on the input side to control large currents on the output side
Often come as sets of 7 or 8 in a single IC package e.g. ULN2003/ULN2008 - as in the project board - often also called a "darlington driver", but actually they are multiple darlington drivers in one package
The most important thing is that they are normally "low side switches":
http://www.opencircuits.com/Basic_Circuits_and_Circuit_Building_Blocks#Transistor_Low_Side_Switch
So they act like a switch in the ground lead of your load (LED, motor, relay etc) and they are like...
...an open circuit/switch when the input to them is low - so LED, motor etc will be off
...a short circuit/closed switch when the input to them is high - so LED, motor etc will be on
They do not "produce" or "source" current themselves, they just switch your load on or off
However, they are not "perfect switches" and have/suffer from having a voltage across them when "on" - maybe 0.6 to 2 volts or so - and in some circumstances you need to be aware of that