Here we can see how the working cycle of a turboprop engine actually compares to that of a piston engine driven propeller.
The inlet receives gas similar to the inlet valve on a cylinder head.
The compressor increases the pressure of the gas, similar to the compression stroke of a cylinder.
The combustion chamber (mixes and) ignites the gas, causing it to expand, similar to the spark induced combustion of a cylinder.
The exhaust allows the expanded gas to escape, similar to the exhaust valve on a cylinder head. The gas turbine extracts energy from the exhaust gas to drive the turbine and consequently, the compressor and propeller.
The simple expression Suck, Squeeze, Bang and Blow is the best way to remember the working cycle of the gas turbine.
The big difference between the 2 types of engine is that a gas turbine employs a continuous cycle, not a 4 stroke cycle, where 3 of the strokes are not actually providing power to the vehicle. This means that more fuel can be burnt in less time, creating more power for a given engine size.