The Darlington pair is basically a combination of two bipolar transistors connected as shown in Figure 1 below.  This circuit is used for amplifying currents, i.e., the amplified current from the first transistor is further amplified by the second transistor. Needless to say, this transistor combination exhibits a much higher current gain than if only one transistor were used.

The overall current gain of the Darlington pair is just equal to the product of the two individual current gains of the transistors. Thus, if hFE1 and hFE2 are the current gains of transistors 1 and 2, respectively, the over-all current gain hFE if they are formed as a Darlington pair will be hFE = hFE1 x hFE2.  The very high current gain (e.g., 10000) of a Darlington pair means that only a tiny amount of base current is needed to make the pair switch on. 

Figure 1.  The Darlington Pair Circuit

The Darlington circuit is named after its inventor, Bell Labs engineer Sidney Darlington. The concept of placing two or three transistors on a single chip was patented by him, but putting an arbitrary number of transistors on a single chip, which would encompass all modern IC's, was not covered by his patent.

   
A Darlington pair behaves just like a single transistor with a very high current gain, i.e., it can also be modeled as having the three terminals of a typical transistor - the base, the emitter, and the collector.  A single transistor requires just 0.7 V across its base-emitter terminal to turn on.  A Darlington pair, on the other hand, requires double that amount, or 1.4 V, since its input basically has two base-emitter junctions in series (refer to Figure 1).  If Vbe1 and Vbe2 are the base-emitter voltages needed to turn on the first and second transistors of the Darlington pair, respectively, then the Vbe required to turn on the pair is Vbe = Vbe1 + Vbe2.

Darlington pairs packaged as a single transistor are already very common in the marketplace. However, this has not diminished the practice of forming Darlington pairs from two discrete transistors, since this offers more flexibility.  The load current will be carried by the second transistor, so it has to have a higher power rating than the first transistor.  The maximum load current of the Darlington pair is the maximum current that the second transistor can carry.