A delay stack is used in a PA system for a larger event in a very long room. In a long room the quality of the sound deteriates as you further down the room as sound is absorbed by people, carpet and furniture. Of course you can add speaker halfway down the room, however sound travels at approx 340 m/s or approx and the sound will be out of time. The sound from your front speakers will have an echo compared to your mid speakers and your event will sound like its a train station!
The solution is to add a small delay to the mid speakers to bring them back in time. I also find it useful sometimes to put a short delay on the main PA if I have a deep stage withe a drummer at the back. Pecussive instruments like snare drum really accentuate timeline errors and a short delay, say 10m/s on the FOH can really tighten up the sound. I tried delaying just the snare channel bbut this affacts the f/back adversely. It is an inexact science as delays change depending whether you are left or right of the PA and how close you are to the PA. But generally delaying sound helps.
The X32 dgital mixer has delayes available on every input and output and is idea for doing this.