Intercooling theory according to BeatFreak:
The idea is that the air enters at high pressure and by PV=nRT, high volume as well. What the intercooler does, is lower the temperature of the air charge. At first, it seems this is a bad idea, since it will lower the pressure as well. And we know this game is all about pressure, right?

As it turns out, not entirely. The name of the game in an internal combustion engine is getting the most oxygen possible into the cylinders to burn what little fuel is used as efficiently as possible.

(By efficient, I mean burning the fuel in such a way that it is consumed completely, and that no molecules are left uncombusted.)

It is this quest that has driven engineers to develop forced induction, which effectively raises the volume of air at the intake manifold waiting to enter the cylinders when the intake valves open, thereby raising the amount of oxygen present during combustion.
See: Turbocharger and supercharger for examples of forced induction in practice.

In such a device, the air gets pressurized to a certain constant point. However, it's also very hot. According to Boyle's Law, when pressure remains constant, and temperature increases, volume must increase as well. This creates an undesirable situation. We have a hot gas, which has a larger volume for a given pressure. Not all of this hot air charge can be used in each intake stroke, and thus creates back pressure, which destroys your turbo or super. Clearly, something has to give. Enter the intercooler.

The intercooler takes the air charge from the device, and exchanges its heat via conduction to the core and fins (like a radiator), and thus by Boyle's Law, as pressure remains constant when temperature decreases, volume must also decrease.

While this seems like an ideal situation, we can't leave out one major fact: As the air cools, it's density increases, but its overall pressure decreases. This phenomenon is known as pressure drop. It's a critical design metric in designing a forced induction system. If your device can't put out enough pressure to make up for the pressure drop accross the intercooler, you won't get any positive pressure at the intake manifold, and thus your device's boost is thoroughly wasted.

Hence, there's a necessary order of operations to follow in designing said system:

1. Measure the driving force your engine will put out to your device.

2. Size your device accordingly, optimizing for low spool time and maximum boost.

3. Size your intercooler according to the pressure output of the device, optimizing for the amount of positive pressure you expect to get at the intake manifold.

To recap: The whole point of forced induction is to increase the amount of oxygen going into the engine. It does this by pressurizing air via some kind of compressor. In the compression process, the air charge heats up which increases the volume (and therefore decreases the density) of the air charge. Which all adds up to less oxygen per stroke. The intercooler significantly cools the air charge, thereby increasing its density, and allowing more oxygen into the engine per stroke. It does this at the cost of decreased pressure. Thus, make sure your device can overcome this.

See intercooler for a practical application.

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