"Long ago, the heralds of pestilence were lesser demons who rebelled against Azmodan. The Lord of Sin vanquished them easily, but instead of slaying them, he granted each one a long arm infested with a poison that could kill anything but Azmodan himself. The heralds have been plague-riddled menaces ever since."— Deckard Cain(src)
Heralds move and attack rather slowly, but have two modes in which to attack.
From afar, they are plunging their oversized arm into the ground, emerging it near the player some time later. Once emerged, the arm starts pumping toxins, dealing Poison damage to everything within roughly 10 yards like a Plagued pool. This attack lasts at least 3 seconds, and Herald cannot do anything else while channeling it. Use it to your advantage to attack the Herald while he is helpless. Being Knockbacked or disabled with the hand in the ground will cause a Herald to stop channeling the attack.
In melee, they just deliver a close combat hit, also dealing a lot of Poison damage and inflicting poison.
Overall, they are more of supporting fighters, as their ranged attack is easily avoidable, but multiple Heralds can infest a large area, blocking much of the path.