Darlington transistor is transistor circuit that is made from two transistors. It is kind of like a transistor but it has much higher ability to gain current. The circuit can be made from two discreet transistors or it can be inside an integrated circuit.
The circuit is useful (for example) in audio amplifiers or in a probe that measures very small current that goes through water. It is so sensitive in fact, that it can pick up the current in skin, so if you connect it to a piece of metal, you can make a touch sensitive button.