Chargers work by compressing the air before it is mixed with fuel inside the cylinder, more air means more oxygen, and more oxygen means a hotter and more energetic fuel combustion leading to increased engine power output (this is the same reason why ICEs have a higher power output in cold weather, because cold air is naturally already more dense than hot air).
The only difference between the two types is how they are driven, a supercharger is mechanically coupled to the ~~driveshaft~~ crankshaft of the engine, a turbocharger (as the name suggests) uses a small turbine that is driven by the engines exhaust.
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