1. Using conventional current flow,
2. Using electron theory,
3. Using resistance analogy, and
4. Using the "visual" approach.

We will cover the 4 different ways and let you work out which is the easiest to understand and how they interact with one-another.

Conventional current flow is easier to understand than electron theory while the resistance approach is more closely allied to the beginners level. But the visual approach is the best. By covering the 4 approaches, everyone will be able to understand the operation.
The author has used super-alpha (Darlington) devices for the past 20 years, when repairing TV's and equipment, and studied the circuits to find out how they operate in each of the applications.
The visual approach is easiest to understand is an NPN transistor with the emitter connected to the negative rail. This is called a COMMON-EMITTER stage. But it works equally well with PNP transistors as you can visualise them as being an upside-down NPN transistor.

This is how to see it:
The BD 679 Darlington transistor is two transistors in one package, connected as shown in the diagram above. The upper transistor is a small-signal transistor and has its collector and base leads coming out of the package. The lower transistor has its collector and emitter leads coming out of the package. Obviously the collector lead is common to both transistors. 
When a voltage is applied to the base of this Darlington device, a current, flows into the base and turns the top transistor ON. Delivering current into the base is exactly the same as lifting the base a small amount with your finger. You don't need very much effort. The base is very easy to lift. This small effort is amplified by the transistor to the emitter and it lifts the emitter with the strength of about 100 times the effort you provided.
This "pulling-up" effect is now transferred to the base of the power transistor and it is equivalent to raising the base with 100 times more force than you exerted on the external base. 
The power transistor inside the device PULLS DOWN very strongly on the bottom of the LOAD RESISTOR and this allows a higher voltage to appear across the LOAD. The result is the load is activated. 
That's it. You lift the base UP very gently and the Darlington transistor pulls DOWN very strongly on the load resistor.

RESISTANCE ANALOGY
Now for the resistance analogy. When no current flows into the base of the transistor, the resistance between the collector and emitter leads is very high. When you allow a small current to enter the base, the resistance between collector and emitter is reduced.
If you allow more current to pass into the base, the resistance between collector and emitter becomes even lower. For example, when no current flows into the base, the collector-emitter resistance may be 10Meg ohms or more.
For a small current into the base, the collector-emitter resistance may be 10k. When more current flows into the base, the collector-emitter resistance may be 100 ohms or as low as a few ohms.
If we have a resistor - called the LOAD RESISTOR - connect to the collector, we create a circuit similar to two resistors in SERIES and current will flow through the load resistor and transistor according to ohms law. As the resistance of the transistor decreases, the current through the pair increases.
There is nothing special about this, it is simple ohms law, as the transistor turns on harder, the current through the load resistor (and the transistor) increases.
If the resistance of the transistor decreases to a few ohms, you can see the current through the load will be a maximum and the device will be activated. 

CONVENTIONAL CURRENT APPROACH
The conventional current approach is the concept that current flows out the positive terminal of a battery and into the negative terminal.
In our case, when a small current flows into the base of the Darlington transistor, the collector-emitter circuit allows a larger current to flow.
When more current is allowed to flow into the base, the transistor will allow a greater current to flow in the collector-emitter circuit.
Since a load resistor is connected to the collector, this collector-emitter current will also flow in the load.

THE ELECTRON APPROACH
The electron approach basically says that electron flow occurs in the opposite direction to conventional current flow. In other words, electrons flow out the base to the positive rail and others flow in the emitter-collector circuit from the negative rail to the positive rail. The electron approach is  necessary if you want to describe the actual operation of transistor itself.

THE VOLTAGE APPROACH
One final way of looking at the operation of the circuit is the VOLTAGE approach.
The transistor does not turn on until a voltage of 0.65v + 0.65v  = 1.3v is supplied to the base lead. 
This voltage allows a small current to flow into the base and the voltage on the collector falls from rail voltage to slightly less than rail voltage. This puts a small voltage across the load and a small current flows in the load. 
As the voltage on the base is increase a few more millivolts, more current is able to flow into the base and the transistor multiplies this flow about 2,500 times and the increased current is allowed to flow in the collector-emitter circuit. 
Increasing the voltage on the base a few more millivolts will allow more current to flow into the base (in the order of milliamps) and very soon the maximum current will flow via the collector-emitter. 
If the load is a relay or globe, the device will be activated. 
You really need to combine all the concepts we have described to see how the transistor works.
Once you can see how the collector and emitter leads "squeeze" together or reduce in resistance, to deliver a current to the load, you will be able to see how the transistor works.

As more current is delivered to the load, this will introduce another feature you need to understand. It's called . . .

TRANSISTOR DISSIPATION
All the current through the load must flow through the transistor - this is obvious as the two are connected in series, and when current flows, heat is produced.
Unfortunately, transistors don't like to get hot and so the heating effect must be kept to a minimum.
When the transistor is off (no base voltage applied) no current flows and thus the heating effect is zero. This is obvious. 
This is one of the states of a transistor and is called "cut-off" or simply "off."
As the transistor gets turned on, more current flows and the heating effect increases.
Do you think this increase will be a maximum when the transistor is fully turned on?
No! Amazingly, the heating effect increases until half rail voltage appears on the collector and starts to reduce until it becomes nearly zero when the transistor is fully turned on. This state is called "saturation" or "bottoming."
This is a very important feature in electronics and means there are two states when the transistor is dissipating the minimum energy (heat) - the two states are: "bottoming" and "cut-off."
When a transistor is used in digital mode, it switches from one state (say cut-off) to the other (saturation) very quickly and very little heat is generated (lost) in the transistor.
But when a transistor is used in an audio application, it moves very slowly between one state and the other and a lot of heat is generated.
This heat must be passed to a heat-sink as quickly as possible to prevent the transistor heating up and self-destructing.
Self destruction may be final or partial and may produce loss of gain, an open circuit or a "short-circuit." Most often a transistor goes short-circuit.

GETTING MORE TECHNICAL
Let's get a little more technical. One of the problems with the BD 679 is the minimum voltage between the collector-emitter terminals, when the transistor is fully turned on. This minimum voltage is about 2-3v, where as a normal transistor can go as low as .3v. For a MOSFET, the voltage can be .02v or lower.
Ideally, this voltage should be as low as possible as it determines the heat lost in the transistor when it is fully turned on. For instance, if a BD 679 is passing 4 amps in its saturated mode, the heat lost will be 2 x 4 = 8watts, or in the worst case, 3 x 4 = 12 watts.
For a MOSFET, the heat loss will be less than 0.08 watts.

WHY?
Why is the minimum collector-emitter voltage of a Darlinglon = 2.5v?
To see why, we need to see the diagram of the transistor as shown in the figure below:

The small-signal part of the Darlington is riding on top of the power transistor. As the collector of the power transistor falls during turn-on conditions, it pulls the collector of the small-signal transistor down with it and robs it of collector-emitter voltage.
This reduces the gain of the small-signal transistor and the collector-emitter current is also reduced so the power transistor will not be able to turn on as hard. This means the voltage on the collector will not fall below 2.5 volts as this leaves only about 2.5v - .65v = 1.85v, for the collector-emitter voltage of the small signal transistor.
This is just enough to give the transistor a collector voltage so that it can provide some gain to drive the power transistor.
This is one of the limitations of a Darlington transistor.

BASE VOLTAGE
Another characteristic of a Darlington transistor is the base voltage. It needs a voltage of 1.3v for the transistor to begin to turn on.
This compares with .65v for a normal transistor.
Darlington transistors consist of two transistors in a staircase arrangement and the base voltage is .65v + .65v = 1.3v. See diagram below:

The two base-emitter voltages combine to produce 1.3v to turn the Darlington transistor ON.

PHOTO DARLINGTON
Other Darlington devices can also be created. A photo-Darlington transistor is available under the part number MEL-12. This is a very sensitive device for detecting light and can be used for many applications such as photo-electric beams, light detection etc. The structure of the MEL-12 is shown below:

MEL-12 Photo Darlington Transistor