After your opponent moves and presses the clock there is a 5 second delay before your time starts counting down.
This is different from increment in that instead of adding 5 seconds, the timer is just delayed from counting down. Both delay and increment allow a player to avoid losing on time by moving quickly, but with delay it's not possible to accumulate any extra time on your clock.
My apologies for asking a question that probably everyone knows about except me, but I played in my first tournament in close to 20 years recently, and the time was 90 minutes with a 5 second delay. I'm not sure what the 5 second delay thing is? Any advice would be helpful