Doppeldeker,
Without spending a lot of time on turbine theory, you can think of a turbojet engine as a device that takes a whole lot of air and moves it into a small space. A turbine engine uses four main parts. One is a compressor to draw the air in and mash a lot of it into a little space. Another is a diffuser which takes the compressed air and causes a pressure rise before dumping it into a burner can. The next is the combustion chamber, which is just what it sounds like...it serves a function somewhat like the fuel injected combustion chamber in a piston engine where fuel and air is mixed and burned...and finally comes the turbine section. The turbine section is where the exhaust gasses go. The burning or burned gasses blow across the turbine blades, which turn a shaft...which moves the compressor. So much for turbine theory...mashing air into a small space.
Ever go through a revolving door? Ever try to go through a revolving door as part of a crowd of people all trying to go through the door? The door mashes up a lot of people into a little space, and the ability of the door to admit a lot of people depends on the folks who pass through doing so in an orderly fashion.
Imagine someone trips, or slips,or bumps into someone else. There's a chain reaction, people slow down, people bunch up at the front of the door. People maybe even get stuck in the door. The chain reaction ripples back and people get shoved and pushed, and find that they can't get through the door. If people find they get trapped in the door they may even react violently, pushing and shoving not inward, but back outside into the cold street. Such is a revolving door, and such is a compressor stall.
To help many turbine engines function properly, special "bleed valves" or "acceleration bleeds" are installed to help prevent air from bunching up or backing up...to help air flow smoothly through the engine. These valves are like little side doors that might open up in the revolving door to allow extra people through...except it's in the engine. These valves are doors or relief ports which will allow some of the air passing through the engine to be vented off until a good, steady flow takes place. They mostly open and close automatically to help "unload" the engine or relieve it of some of the air flowing through, to prevent it from backing up or...you guessed it...stalling.
As others have indicated, a stall is nothing more than airflow exceeding a critical angle of attack of an airfoil, and it can happen as easily in a turbine engine full of little airfoils as it can to a wing or rotor or propeller...a stall is a stall. A compressor stall is more than just an issue with angle-of attack of individual blades, however, and is a problem with the overall airflow through the engine (particularly the compressor)...and may stem from multiple causes ranging from a sticky acceleration bleed to a dirty compressor to too high an engine demand with too low airflow, too high an angle of inlet airflow, a birdstrike, or anything else that adversely alters airflow through the engine.
That's what causes it. What it is from the cockpit is something else. It may be bouncing needles on a cockpit indication. It may be a low, subtle hooting noise like an asthmatic beagle begging to be let in, or it may bellow out there like an upset walrus. It can bang away like nobody's business and sound like shotguns going off, or simply chug and vibrate and shake. It may be subtle, or may be something that can't be ignored.
Something like this, though for very different causes, can take place in the engine of your Cessna 152. This doesn't happen because of the propeller, but for other reasons that can range from a slipping magneto to a sudden change in engine operation. You can get backfiring, through the induction, or afterfiring, through the exhuast. The closest you might get in the piston engine to a compressor stall is a backfire...which can bark out through your engine intake and can damage your carburetor, carb air box, air filter, or induction.