It's usually audibly a pop, bang, or buzzing sound. It will sometimes (usually?) be accompanied by some sort of change in RPM (usually decreasing) or ITT/EGT/FTIT (usually increasing). Visually, it may be accompanied by some smoke or a flash out either end.
I've had three compressor stalls; one in the F-15E and two in the T-38. In the F-15E (PW-220 engine) it was a big bang (like a gunshot) with a 15-foot flame shooting out the intake for a second. There was a quick jump in FTIT, but everything else stayed the same in terms of engine indications. There was a slight lurch simultaneous to the 'bang', but no other noted loss of thrust. The engine recovered and had normal performance throughout the envelope.
Two very different stalls in the T-38. One was a small pop, followed by a puff of smoke out the exhaust, and then a constant buzzing sound (and vibration through the airplane). RPM slowly started to decrease and EGT inreased rapidly. The engine quit, and I did not attempt to restart it.
A second time was a violent buzz as I pushed the throttles into afterburner, with both decreasing RPM and EGT. I ripped the throttle back to idle (IAW the checklist non-memory memory item) and the engine recovered at idle.
EDIT: Forgot to add: of note here is that the Pratt 220 engine is controlled by a DEEC, and the GE-5 engine in the T-38 has a manual fuel control. There is a direct correlation between the incidences of compressor stalls and the use of a digital engine/fuel control (ergo, there are less compressor stalls with DEECs/FADECs than without).