Effect
Hot Pursuit
Hot Pursuit is a passive that increases your movement speed for a few seconds as soon as you hit an enemy. Since you attack essentially nonstop in combat, the speed bonus is almost permanent in a fight and makes the Demon Hunter noticeably nimbler at kiting, dodging, and repositioning. For a class that lives on agility and range, that's a pleasant, low-threshold advantage.
The passive is valuable above all in fast, mobile playstyles, such as speed-farming lower content, where pace translates directly into efficiency. The extra speed also helps keep distance when kiting dangerous enemies. In pure burst or stationary-damage builds where you barely move, or when your speed is already capped elsewhere, its merits recede into the background.
Keeping the bonus running
Every hit on an enemy grants 20% movement speed for 4 seconds, so as long as you hit something at least every 4 seconds, the bonus stays up. It ends once you stop hitting for longer, for example on the way between two packs or while you wait behind cover, and that is exactly where you lose the speed. The passive unlocks at level 27, so you can use it early while leveling.
Two speed passives with different triggers
Tactical Advantage often sits next to it and complements Hot Pursuit because it triggers differently: not on hits but on Vault, Shadow Power, Smoke Screen or the backflip from Evasive Fire. That gives you a burst of speed even when you are not hitting anything and need to escape a threat.
Discipline for Smoke Screen and Shadow Power
Smoke Screen and Shadow Power cost Discipline; Preparation refills it instantly, and Blood Vengeance makes health globes restore Hatred and Discipline as well. Vengeance raises your damage in the meantime, and the faster enemies drop, the sooner you reach the next pack, where Hot Pursuit kicks in again.