Sometimes you want to control a really large amount of current; more than a single transistor's gain can provide. I.e. the output current has to be more than the input current multiplied by the transistor's gain.
In this case we can combine two transistors in what is called a Darlingtion connection. In this configuration the gain of the pair is very slightly more than the product of the gain on the individual transistors and can be 1000 or more. Specifically, its the product of the gains plus the sum of the gains, but the product is usually high enough that the sum can be ignored for calculations.
At its core, a Darlington pair is simply two transistors cascaded together:
A common Darlington pair is the TIP120 which is a great choice for the examples later in this guide. It can handle up to 5A continuously (with bursts of 8A) and up to 60V across the collector and emitter. It has a gain of ~1000.
A TIP120 is just the circuit above packaged in a convenient 3-pin package (base, collector, and emitter). Since it's meant for power switching/amplification it comes with a metal backplate to which a larger heatsink can be attached.
The "drive" connections of the CRICKIT are, in fact, darlington drivers. They are all on a single IC, called a ULN2803, with 7 or 8 Darlington's each with the following design:
This chip is very convenient to use as it provides everything needed: all the biasing resistors, and all the protection diodes needed for inductive loads such as electromagnets, solenoids, and relays.
The CRICKIT uses 4 of these drivers, though the chip contains 7. There are a variety of versions of this design, some with 8 drivers in a slightly larger chip.