The Turboprop engine is made to run the same way by which the turbojet and turbofan engines work. The incoming air is compressing air with the compressor blades, and forcing it into the ignition chamber. The exhaust of the combustion forces the rear turbine blades to turn, which in turn moves the common shaft for the compressor blades and the ignition chamber.
The only difference is that the shaft's main purpose is to move the propellors in the front of the engine, which provides the thrust to the plane. Though, it doesn't provide a high thrust, it is fairly economical as compared to the other two engines, and is used mainly in transports.