Generally speaking:
propellers have better efficiency at lower speeds, but limit your top speed, because you get issues with the blade tips reaching the speed of sound otherwise. This would be really loud, and have bad efficiency.
Modern jet engines are Turbo Fans, which are basically a duct around the propeller. They slow the air down before it enters the large Fan (we call the propeller Fan now) (the diameter at the intake is smaller than the Fan diameter. This Expansion slows the air down), which allows the to encounter this speed of sound issue later. (the tips of the blades are however still supersonnic, but because they are in the duct, this isn’t that loud, and produces fewer efficiency issues). Jets can thus fly at higher speeds.
Both engine types are however powerd by the same thing (in larger aircraft, with let’s say more than 600 hp per engine), a turbine. On small aircraft, normal piston engines are used.
Right now, several companies are trying to develop a mixture between a propeller and fan (again), the propfan. This might be used on the future version/replacement of the 737 and a320.
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