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For the sake of this answer, we will divide all the airplanes in the world into 2 classes. Jet, and non-jet (uses a Propeller).

Jet engines suck air from the front, compress it to around 40 times it's original volume, ignite it, and then shoot it from the back, the engine exerts a huge amount of force on the air escaping from behind, which in turn exerts the same amount of force on the engine, which causes it to accelerate forward. Since the engine is firmly anchored to the airplane, the entire plane accelerates forward.

As for propeller based aircrafts, the propeller works the opposite then your household fan. instead of taking air from the back and "shoot" it forward, the propeller takes air from the front, and shoots it backwards, which incredible force. this causes the same action-reaction seen on a jet plane, only on smaller scale.

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