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ElI5: turbocharging as it relates to automobiles.

How does it work, what function do all the different components (blow-off valve, dump tube, wastegate, intercooler, hot side/cold side etc.) perform?
gpt
In a normal NA engine, exhaust gasses are sent straight down the exhaust pipe and are not used.

In a turbo engine, exhaust gasses are re-routed back into the turbo-charger. The exhaust gasses spin up the charger which compresses the air which is then sent into the engine. Because the air is compressed, it makes a bigger bang, which in turn makes more power.

There are two sides to the turbo, the hot side and the cold side. The hot side is the bit which the exhaust is piped into - this is the gas which is used to spin the turbine to compress the air.  The cold side is the bit which is drawing in the cold air which is sent into the engine to make the bang.

The cold side will be plumbed into either a standard air filter or into an inter-cooler. If it is plumbed into the standard air filter then it will breath regular air. However, if you want to make more power, you can plumb in into an inter-cooler which means it will be taking in cooled air. As cooled air is more dense, there is more of it in a given area, thus it makes a bigger bang which makes more power. Out of the factory, most turbo cars have an inter-cooler. However, many people who want to modify their NA cars to add a turbo often skip the inter-cooler due to space limitations in the engine bay or to save a bit of money.

The turbo needs a mechanism to regulate the amount of compression. If you have too much, it can put more air into your engine than its designed to manage and cause serious damage. Therefore the turbo includes various mechanisms to regulate the boost pressure. The wastegate is an example of this, its purpose is to act as a control valve so that should the pressure reach more than your pre-defiend acceptable level, it will enable excess air to escape thus keeping a managable level of boost. The wastegate sits in the pipework between the engine and the hot side of the turbo and can release excess pressure before it gets to the turbo. 

Many people who modify their turbo cars to make more power do so by stopping the point at which the wastegate dumps air - this increases the boost pressure beyond the manufactures level and is an easy way to make more power. The downside is that it comes at a risk of if you set it too high, the engine blows up.

A blow off valve / dump valve (same thing) sits on the pipework leading to the cold side of the turbo. This is often confused with the wastegate but its a different thing as its on the cold side rather than hot side. Its job is to dump any excess pressure when the driver lifts off the throttle (eg gear changes) to ensure that any back-pressure which is built up which could mess with the turbos compressor is dumped rather than goes into the turbo. Some dump valves are pretty much silent and they re-circulate this air back into the system, others make a 'tsschhhh' type sound which is when the compressed air is being dumped into the atmosphere. Most cars come as standard with the silent recirculation type but its a fairly easy modification to swap it too the louder one if thats your thing.