The spinning blades in the compressor stage of a jet engine or turbo fan are themselves airfoils, like the aircraft's wings. The operation of the engine depends on the smooth flow of air over the blades.
Just like a wing, an individual blade, or a small component of one, can experience an airfoil "stall", where the air flow over the blade separates into a cell of "stuck", highly turbulent air behind the blade, and the air flows around the cell instead of smoothly around the blade.
When a flow separation occurs, the airfoil's ability to push the air in proper direction at the right flow rate—and thereby contribute to the compression of air behind the fan assembly—is inhibited.
Since these blades are rotating, the blade quickly moves away from the packet of stagnant air. Of course the stalled air packet has some momentum from the intake air, and it will experience a drift along with the rotating blade, but it is not moving with the average flow of the air through the engine anymore.
Thus the next blade spinning round tends to encounter the stalled air packet. If the stalled air cell is not particularly large, it may be absorbed by the air flow at this stage and dissipated. Alternatively, it might be large enough to stall the subsequent blade as well. At this point, it is called a "rotational stall".
If the stall continues to propagate, the ability of the fan stage to deliver air to the subsequent compression stage is impeded, and it will lead to an abrupt drop in pressure inside the combustion chamber. This causes a reduction in available oxygen for combustion. The engine's performance, measured by the thrust delivered, is strongly impaired, and there is likely to be a lot of unburnt fuel remaining after the oxygen in the compressor is exhausted. That unburnt fuel may ignite in a bright exterior flame as it escapes out the back of the combustion chamber and mixes with the oxygen-rich bypass flow, or in the case of a jet engine, after it exits the engine altogether.
The drop in back pressure in the compressor will, under normal conditions, enable the compressor fan blades to begin operating as proper air foils again.
An alternative type of stall is a compressor surge. In this case, the problem is caused by unexpectedly high pressure in the combustion chamber (or in the compressor), which forces its way forward as well as backwards, against the incoming flow rate. Again, this disrupts the operation of the compressor blades as airfoils.
See Advanced Control of Turbofan Engines by Hanz Richte