I have a vague idea that air compression involved but I don’t know how this fits together in the grand scheme of things.
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A jet engine works by sucking in air through a series of fans, compressing it, mixing it with fuel, and igniting it. The hot exhaust gases then exit out the rear of the engine, passing through more fans (the turbine) that in turn drives the compressor fans at the front via a shaft.
The hot exhaust gases create pressure that drives the engine (and anything attached) forward.
A rocket engine works by mixing fuel + and oxidizer to create a chemical reaction, which in turn creates hot exhaust. The major difference between a Jet Engine and a Rocket is that the Rocket has to bring it’s oxidizer (air supply) along with it, while the Jet engine sucks in air from the outside.
A propeller meanwhile is a number of smaller airplane wings that push air as it spins. So long as a propeller is spinning it pushes air and generates thrust.
Propellers can be spun by a number of things like piston engines, or even jet engines. A Jet engine with a propeller attached is called a turbo-prop, and it works by attaching the turbine (the fans in the exhaust) to a drive shaft and gearbox that turns the propeller.
By why use a turbo prop instead of a jet engine? These engines are slower, but much more fuel efficient than jets making them ideal for smaller commuter aircraft. The turbines are also much lighter than piston engines creating much more power to weight.
A propeller is a set of wings turned horizontal and spinning around a shaft. These wings create lift which when turned sideways is instead thrust.
A jet engine is a turbine system that has a big fan in the front to intake air and direct it to more fans in the center of the engine that squeeze this air down until it is compressed to a correct level. That air then has fuel injected into it and it is combusted. Combustion creates an expansion and a lot of pressure. That pressure is forced out the back of the engine through another fan blade that is connected to the ones at the front and powers them. This exhaust forced out the back at high speed creates thrust. (Same as a water hose whipping around if you don’t hold onto it.)
A rocket engine is like the rear part of a jet engine without as many fans. Rockets have to work in space, so they carry their fuel and their oxygen along. These get mixed in the rocket engine and ignited. Same result as a jet engine, the pressure of the combustion is directed out the rear via the nozzle and that force is thrust. The shape of the nozzle helps make the engine work better in different environments (such as sea level for launch, or in the vacuum of space.)
Propeller doesn’t require combustion it just pushes air, same as a propeller in a boat. Rockets don’t require oxygen inlets, typically the fuel has it’s own oxygen for combustion.
A jet turbine uses the fans to feed air in and compress it then combusts the fuel in the compressed air to create thrust. Some jets, most commercial, just use the combustion as a means of driving more fans. But high performance jets, think miltary jets like the f16, can use the combustion in afterburners which makes the engine run more similar to a rocket. All are pretty similar but slight variations.
Propeller engines aren’t actually “just” propellers; the important part is what actually drives the propeller, and for larger commercial planes propellers are driven by turbine engines that aren’t that different from jet turbines.
In any case, regardless of whether it’s a turbine or a reciprocating (i.e. like the one in a car) engine, they function by taking in air from the atmosphere, adding a little fuel, compressing the mixture, igniting it, and controlling how the gas expands so as to create rotational power and/or thrust.
Rockets work completely differently as they have their own sources of fuel *and* oxidizer on-board (there’s no oxygen in space, so if you want combustion you have to bring it with you). For rockets, fuel and oxidizer (together, propellant) are mixed and ignited and then expanded at extremely high velocities through a nozzle, which is then pointed in the opposite direction that you want to go in order to provide thrust. There’s no real control in the expansion beyond vectoring the thrust, whereas in other combustion engines you typically get some energy back at the expansion phase that you use to run other processes (i.e. driving the compressor fans in the turbine, keeping the engine running between combustion cycles in reciprocating engines, driving smaller generators that provide electrical and hydraulic power for both, etc.).
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