system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: the difference between a turbocharger and a supercharger.
gpt
A turbocharger has two halves -- one half is connected to the exhaust system, and one half to the intake system.  Air passes through both halves during operation, but never from one side to the other directly.  So when your engine burns the air/fuel mixture, it produces an exhaust gas.  This is hooked up to one half of the turbo charger.  As the exhaust gas passes through the turbo, it spins a turbine (fan) and then exits out the back of your car.  

That fan that's spun by the exhaust gas leaving the engine has a counter part fan on the intake side, linked with a shaft.  In other words, when the exhaust side fan spins, it also spins the intake side fan.  The intake side does the opposite -- it pulls in outside air and then forces it into the motor.  With more air, you can put in more fuel, and get more power.

A supercharger skips the exhaust side of the turbo.  Instead, the compression mechanism is spun by a belt connected to the crankshaft (the shaft that essentially spins the wheels).  So as the engine revs up, it spins the belt, which in turn, spins the supercharger connected to it.  This draws in air then forces it in to the motor.  

Differences:

A turbo has lag.  It takes time to spin up and get the airflow going.  In other words, you step on the gas and have to wait for the turbo to spool up to get the power.  Now there are different technologies that help alleviate lag, but it's kind of the defining aspect of a turbocharged car.  On the other hand, it's more efficient -- since it uses the natural byproduct of combustion (the exhaust gasses), it doesn't really use any extra power.  Last, turbos generally have the capability of putting out higher levels of boost -- but this is also determined by the size of the turbo. 

A supercharger has no lag.  You push the gas, you're immediately getting the benefit.   However, superchargers aren't as efficient -- they have a slight parasitic (power draining) effect because it uses some of the power being produced by the motor to spin the turbines.  In big power applications they tend to not be able to produce as much boost, but it's usually a moot point.