NextDelay is a property that activates a hidden "hit timer" that limits how often any attack from that group can damage a single target. This works similarly to the infamous spell timer, which limits how often any spell in that group can be cast. The difference is that the hit timer doesn't prevent you from attacking, it just sets everyone's chance-to-hit that monster (with a hit timer attack) at zero until the timer runs out.

Here's a list of skills (Assassin skills in italics) that are affected by NextDelay, and the lengths of them:

25-Frame Delay (1 second)

Shock Web
Blade Sentinel
Twister
Tornado
10-Frame Delay (0.4 seconds)

Volcano (initial eruption)
6-Frame Delay (0.24 seconds)

War Cry
Grim Ward
5-Frame Delay (0.2 seconds)

Fissure
Volcano (small fireball)
4-Frame Delay (0.16 seconds)

Multi Shot
Strafe
Lightning Strike
Chain Lightning
Nova
Frost Nova
Poison Nova
Shock Wave
FoH Holy Bolts
Wake of Fire
Claws of Thunder
Phoenix Strike
Dragon Flight
Battle Cry
Battle Command
Battle Orders
So, the moment a certain skill from the list damages a monster, it prevents every other skill on the list from hitting the said monster for the original skill's NextDelay duration (whether from you or any other character).